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Limite et continuit\303\251 d'une fonction \303\240 une variable\302\251 Pierre LantagneEnseignant retrait\303\251 du Coll\303\250ge de MaisonneuveLa premi\303\250re version de ce document est parue \303\240 l'automne 1998. Ce document actualis\303\251 renferme des \303\251l\303\251ments de programmation Maple qui ne sont pas destin\303\251s \303\240 l'apprentissage de Maple par l'\303\251l\303\250ve. Ces \303\251l\303\251ments sont, par contre, utile \303\240 la pr\303\251sentation des concepts math\303\251matiques de la limite et de la continuit\303\251.Bonne lecture \303\240 tous !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 Ce document Maple est ex\303\251cutable avec la version 2020.1Initialisation restart;with(plots,setoptions):
setoptions(size=[300,300],axesfont=[times,roman,8],color=navy ):with(Student[Calculus1]):L'initialisation suivante permettra d'avoir plus de lisibilit\303\251 des nombres d\303\251cimaux en supprimamt les z\303\251ros non significatifs \303\240 la fin d'un nombre. interface(typesetting=extended); # Pour s'assurer le niveau de composition \303\251tendue
Typesetting:-Settings(striptrailing=true);Digits:=20;Pr\303\251sentation inituitive de la notion de limiteSaississons la fonction f d\303\251finie par 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.f:=x->(x^3-4*x^2+3*x)/(x-3);Tra\303\247ons le graphique de cette fonction .plot([x,f(x),x=-3..4]);Sur le graphique ci-haut, il semble que la fonction f soit une fonction d\303\251finie partout sur LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYmLUYjNiktSSNtaUdGJDYjUSFGJy1GIzYlLUkjbW9HRiQ2LVEoJm1pbnVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JFEiM0YnRjpGOi1GNzYtUSIsRidGOkY9L0ZBUSV0cnVlRidGQkZERkZGSEZKL0ZNUSYwLjBlbUYnL0ZQUSwwLjMzMzMzMzNlbUYnLUZSNiRRIjRGJ0Y6LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y/RjpGOi8lJW9wZW5HUScmbHNxYjtGJy8lJmNsb3NlR1EnJnJzcWI7RidGW29GXm9GOg==. On n'observe aucun trou, aucune coupure.Calculons f(3):f(3);On pouvait s'y attendre puisque le domaine de cette fonction rationnelle est \342\204\235\342\210\226{3}. Analytiquement, examinons les points de la fonction f \303\240 l'aide des valeurs d'abcisses x voisines de la valeur 3. Pour commencer, construisons une suite de nombres voisins de 3 mais toujours inf\303\251rieurs \303\240 la valeur 3 de telle mani\303\250re que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNigtRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEiM0YnRkFGQUZBL0krbXNlbWFudGljc0dGJFEkYWJzRicvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR0ZqbkZlbi8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGRUZBLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEtJnJpZ2h0YXJyb3c7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkQtSSNtbkdGJDYkUSIwRidGLy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJ0Yv . On dit que ce sont des voisins \303\240 gauche de 3: formule:=3-10^(-i);
suite:=[seq(formule,i=1..8)];Maple op\303\250re tr\303\250s bien l'arithm\303\251tique des nombres rationnels sous la forme de fractions. Mais pour notre propos, op\303\251rons les nombres rationnels sous leur \303\251criture d\303\251cimale.abscisses:=evalf(suite);Dans cette liste limit\303\251e, il est clair que 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.Obtenons maintenant les valeurs des images de ces voisins de 3.ordonn\303\251es:= map(f,abscisses);Voici donc la liste des points obtenus.points:=zip((x,y)->[x,y],abscisses,ordonn\303\251es);On constate ceci: lorsque la valeur de x est de plus en plus voisine de 3, tout en \303\251tant inf\303\251rieure \303\240 3 o\303\271 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNigtRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEiM0YnRkFGQUZBL0krbXNlbWFudGljc0dGJFEkYWJzRicvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR0ZqbkZlbi8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGRUZBLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEtJnJpZ2h0YXJyb3c7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkQtSSNtbkdGJDYkUSIwRidGLy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJ0Yv, la valeur des images de ces voisins est de plus en plus proche de 6. On utilise la notation math\303\251matique suivante pour exprimer ce fait.Limit(f(x),x=3,left)=limit(f(x),x=3,left);REMARQUE: le signe moins (LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVEoJm1pbnVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZA) apparaissant \303\240 c\303\264t\303\251 du nombre 3 signifie que les voisins de 3 sont inf\303\251rieurs \303\240 3 avec 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. Ces voisins imm\303\251diats ne sont pas des nombres n\303\251gatifs.Dans le m\303\252me ordre d'id\303\251e, mais cette fois-ci avec des valeurs de x voisines \303\240 droite de 3.formule:=3+10^(-i);
suite:=[seq(formule,i=1..8)];
abscisses:=evalf(suite);Obtenons maintenant les valeurs des images de ces voisins de 3.ordonn\303\251es:= map(f,abscisses);Voici donc la liste des points obtenus.points:=zip((x,y)->[x,y],abscisses,ordonn\303\251es);On constate ceci: lorsque la valeur de x est de plus en plus voisine de 3, tout en \303\251tant sup\303\251rieure \303\240 3 o\303\271 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, la valeur des images de ces voisins de 3 est de plus en plus proche de 6. On utilise la notation math\303\251matique suivante pour exprimer ce fait;Limit(f(x),x=3,right)=limit(f(x),x=3,right);Lorsque des voisins d'un nombre a, incluant les voisins \303\240 gauche de a et \303\240 sa droite, ont leur image de plus en plus voisine du m\303\252me nombre L, on utilise la notation de limite en omettant le signe + et le signe -.Limit(f(x),x=3)=limit(f(x),x=3);En r\303\251sum\303\251, dans la macro-commande limit, si la \302\253 direction \302\273 n'est pas sp\303\251cifi\303\251e et si l'\303\251valuateur donne pour r\303\251sultat un nombre r\303\251el, on peut affirmer donc que la limite existe et vaut ce nombre.La limite d'une fonction f en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lK2V4ZWN1dGFibGVHRkJGPg== existe si et seulement si pour tout voisinage de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lK2V4ZWN1dGFibGVHRkJGPg==, 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 = 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 o\303\271 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZpc2ludjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSgmIzg0Nzc7RicvRjBGPUY5LyUrZXhlY3V0YWJsZUdGPUY5.REMARQUE: Dans l'\303\251valuation d'une limite en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lK2V4ZWN1dGFibGVHRkJGPg== d'une fonction r\303\251elle \303\240 valeurs r\303\251elles, il n'y a que deux approches directionnelles vers x. Et lorsqu'une de ces limites "directionnelles" existe et vaut LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZpc2ludjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSgmIzg0Nzc7RicvRjBGPUY5LyUrZXhlY3V0YWJsZUdGPUY5, la d\303\251finition m\303\252me de la limite ne pr\303\251cise en rien de la mani\303\250re dont les images de f s'pprochent de L. Traitons maintenant un exemple de calcul de limite o\303\271 la limite demand\303\251e en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lK2V4ZWN1dGFibGVHRkJGPg== n'existe pas bien que les limites \303\240 gauche et \303\240 droite existent toutes deux.Soit la fonction g d\303\251finie par 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. Clairement, le domaine de la fonction g est \342\204\235\342\210\226LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNiYtRiM2KC1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1ErWzAsMTYwLDgwXUYnLyUscGxhY2Vob2xkZXJHRj0vJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRj0vRjlRJ2l0YWxpY0YnRjgvJSVvcGVuR1EpJmxicmFjZTtGJy8lJmNsb3NlR1EpJnJicmFjZTtGJ0Y4. Calculons d'abord la limite \303\240 gauche en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjBGJ0Y+Rj5GKy8lK2V4ZWN1dGFibGVHRkJGPg==.g:=x->abs(x)/x;
Limit(g(x),x=0,left)=limit(g(x),x=0,left);Calculons ensuite la limite \303\240 droite en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjBGJ0Y+Rj5GKy8lK2V4ZWN1dGFibGVHRkJGPg==.g:=x->abs(x)/x;
Limit(g(x),x=0,right)=limit(g(x),x=0,right);Puisque 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, la limite 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 n'existe pas.Observons comment l'\303\251valuateur traitera la requ\303\252te du calcul de la limite en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZlcXVhbHM7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOS1GLDYjUSFGJy8lK2V4ZWN1dGFibGVHRj1GOQ== sans sp\303\251cifier la \302\253 direction \302\273.Limit(g(x),x=0)=limit(g(x),x=0);L'\303\251valuateur r\303\251pond, \303\240 sa mani\303\250re, que la limite demand\303\251e n'existe pas. Et nous savons pourquoi.Tra\303\247ons le graphique de la fonction g sur l'intervalle 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.plot([x,g(x),x=-5..5],discont=true);Voici un dernier exemple de calcul de limite avant que nous n'abordions le concept de la continuit\303\251 en un point.
Soit la fonction g d\303\251finie par 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. Pour que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJ4RidGNEY3LyUrZXhlY3V0YWJsZUdGQkY+Rj5GPkYrRmZuRj4= puisse \303\252tre d\303\251finie, il faut, d'une part, que le calcul de 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 et d'autre part, que 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. Il faut donc que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSgmbWludXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrLyUrZXhlY3V0YWJsZUdGPkY1 > 0 Le domaine de la fonction g est donc 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. Cela suffit pour conclure que la LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1JJ211bmRlckdGJDYlLUknbXN0eWxlR0YkNictSSNtb0dGJDYxUSRsaW1GJy8lJXNpemVHUSMxMkYnLyUrZm9yZWdyb3VuZEdRLlsxNDQsMTQ0LDE0NF1GJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdGQy8lKnNlcGFyYXRvckdGQy8lKXN0cmV0Y2h5R0ZDLyUqc3ltbWV0cmljR0ZDLyUobGFyZ2VvcEdGQy8lLm1vdmFibGVsaW1pdHNHUSV0cnVlRicvJSdhY2NlbnRHRkMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4xNjY2NjY3ZW1GJ0Y7Rj5GQUZELUYjNiYtRiw2JVEieEYnLyUnaXRhbGljR0ZWL0ZFUSdpdGFsaWNGJy1GODYtUScmcmFycjtGJ0ZERkpGTC9GT0ZWRlBGUi9GVUZDRlcvRlpRLDAuMjc3Nzc3OGVtRicvRmduRmhvLUkjbW5HRiQ2JFEiMUYnRkRGRC8lLGFjY2VudHVuZGVyR0ZDLUY4Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkRGSkZMRk5GUEZSRmZvRldGWS9GZ25GZW4tRiM2Ji1GLDYlUSJnRidGXm9GYG9GYHAtSShtZmVuY2VkR0YkNiQtRiM2JUZbb0ZBRkRGREZERitGREYrRkFGRA== n'existe pas puisque 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 n'existe pas. (Il n'y a aucun voisinage \303\240 droite de 1 pour approcher 1) Qu'en est-il de la limite \303\240 gauche 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?g:=x->(sqrt(1-x)+x)/(1-x);
Limit(g(x),x=1,left)=limit(g(x),x=1,left);La limite 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 n'existe pas en d\303\251pit du signe \303\251galit\303\251: la notation \342\210\236 est un concept et non pas un nombre. J'aimais bien dire \303\240 mes \303\251tudiants que le symbole \342\210\236 est une vue de l'esprit et si on d\303\251battait s'il s'agissait d'un nombre, je leur demandais quel est-il ? L'utilisation du signe \303\251gal est seulement pour nous informer que lorsque x prend des valeurs de plus en plus voisines de 1, tout en \303\251tant inf\303\251rieures \303\240 1 et tel que 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 , les images de ces voisines deviennent, quant \303\240 elles, de plus en plus grandes et ne se rapproche pas d'un certain nombre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZpc2ludjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSgmIzg0Nzc7RicvRjBGPUY5LyUrZXhlY3V0YWJsZUdGPUY5.Tra\303\247ons donc le graphique de la fonction g sur l'intervalle ]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[ "pour voir".plot([x,g(x),x=-5..1]);Bref, pour LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJSZsdDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GM1Enbm9ybWFsRidGQ0Y9RkBGQw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEtJnJpZ2h0YXJyb3c7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkQtRiw2LVEoJmluZmluO0YnRi9GMkY1L0Y4RjRGOkY8Rj5GQC9GQ1EmMC4wZW1GJy9GRkZMLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJ0Yv 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEtJnJpZ2h0YXJyb3c7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkQtSSNtbkdGJDYkUSIwRidGLy8lK2V4ZWN1dGFibGVHRjQvJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RidGLw==.Plus loin dans ce document, nous nous pr\303\251occuperons de la mani\303\250re dont LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GM1Enbm9ybWFsRidGQ0Y9RkBGQw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEtJnJpZ2h0YXJyb3c7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkQtRiw2LVEoJmluZmluO0YnRi9GMkY1L0Y4RjRGOkY8Rj5GQC9GQ1EmMC4wZW1GJy9GRkZMLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJ0Yv.D\303\251finition de la continuit\303\251 en x = a d'une fonction fLa continuit\303\251 est une notion importante et complexe en math\303\251matique. C'est une notion se situant au coeur du d\303\251veloppement du Calcul. On peut pr\303\251senter intituivement la notion de la continuit\303\251 d'une fonction de mani\303\250re g\303\251om\303\251trique comme ceci: une fonction continue en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ est celle dont le graphique ne pr\303\251sente aucune interruption LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+, aucun trou 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 aucun saut LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+, aucune asymptote verticale LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+. Gardons tout de m\303\252me \303\240 l'esprit que cette mani\303\250re g\303\251om\303\251trique de caract\303\251riser la notion de la continuit\303\251 est imparfaite. Cette pr\303\251sentation peu formelle est reprise plus rigoureusement de mani\303\250re analytique comme suit:Une fonction f est dite continue en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZXRlpGPg== si et seulement si 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. La v\303\251rification de cette \303\251galit\303\251 comporte trois \303\251l\303\251ments afin de montrer que l'\303\251galit\303\251 est satisfaite: 1) Il faut que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJhRidGNEY3LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GPkZmbkZpbkY+RitGZm5GaW5GPg== soit d\303\251finie, autement dit que 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 2) Il faut que 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 existe 3) Il faut que 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On a aussi la d\303\251finition suivante: Une fonction f est dite continue en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZXRlpGPg== si et seulement si 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Par d\303\251finition donc, si l'\303\251galit\303\251 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 n'est pas v\303\251rifi\303\251e, la fonction f n'est pas continue en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZXRlpGPg==. On dit aussi de la fonction f qu'elle est discontinue en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZXRlpGPg==.Afin d'\303\252tre en mesure de d\303\251finir la continuit\303\251 d'une fonction f sur un intervalle 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, il est n\303\251cessaire de d\303\251finir la continuit\303\251 \303\240 gauche et la continuit\303\251 \303\240 droite de f en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+. Cela d\303\251coule de l'existence des limites directionnelles de f en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+.Une fonction f est dite continue \303\240 gauche en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZXRlpGPg== si et seulement si 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Une fonction f est dite continue \303\240 droite en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZXRlpGPg== si et seulement si LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KkYrLUYjNiktSSdtdW5kZXJHRiQ2JS1JJ21zdHlsZUdGJDYnLUkjbW9HRiQ2MVEkbGltRicvJSVzaXplR1EjMTJGJy8lK2ZvcmVncm91bmRHUS5bMTQ0LDE0NCwxNDRdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvSSttc2VtYW50aWNzR0YkUSZpbmVydEYnLyUmZmVuY2VHRkUvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRidGPUZARkNGRi1GIzYpLUYsNiVRInhGJy8lJ2l0YWxpY0dGWC9GR1EnaXRhbGljRictRjo2LVEnJnJhcnI7RidGRkZMRk4vRlFGWEZSRlQvRldGRUZZL0ZmblEsMC4yNzc3Nzc4ZW1GJy9GaW5Gam8tRiw2JVEiYUYnRmBvRmJvLUY6Ni1RIitGJ0ZGRkxGTkZQRlJGVEZob0ZZL0ZmblEsMC4yMjIyMjIyZW1GJy9GaW5GY3BGPUZDRkYvJSxhY2NlbnR1bmRlckdGRS1GOjYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZGRkxGTkZQRlJGVEZob0ZZRmVuL0ZpbkZnbi1GIzYoLUYsNiVRImZGJ0Zgb0Zib0ZncC1JKG1mZW5jZWRHRiQ2JC1GIzYmRl1vRj1GQ0ZGRkZGPUZDRkZGK0Y9RkNGRi1GOjYtUSkmZXF1YWxzO0YnRkZGTEZORlBGUkZURmhvRllGaW9GW3AtRiM2KEZdcUZncC1GYXE2JC1GIzYmRlxwRj1GQ0ZGRkZGPUZDRkZGK0Y9RkNGRkYrRj1GQ0ZGDe ces deux d\303\251finitions, on peut affirmer qu'une fonction f est continue en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ si et seulement si la fonction f est, \303\240 la fois, continue \303\240 gauche et \303\240 droite en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+.Une fonction f est continue sur un intervalle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZzdWJzZXRlcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSRkb21GJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJmRidGL0YyRjlGOS1GLDYlUSRxdWVGJy9GMEY9RjktRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZcb0Y5si elle est continue 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. Lorsqu'il s'agit de la continuit\303\251 aux bornes de I, il s'agit de la continuit\303\251 \303\240 gauche ou \303\240 droite, selon les cas o\303\271 les extr\303\251mit\303\251s de I sont incluses ou non dans I. C'est le cas si I est ferm\303\251 ou semi-ferm\303\251: 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, 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, 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.La discontinuit\303\251 d'une fonction f en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ peut \303\252tre caract\303\251ris\303\251e de deux fa\303\247ons: discontinuit\303\251 essentielle et discontinuit\303\251 non essentielle.cette discontinuit\303\251 en 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 est essentielle si 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 n'existe pascette discontinuit\303\251 en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJhRidGNEY3Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ est non essentielle si 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 existeTra\303\247ons, de nouveau, la fonction de la section pr\303\251c\303\251dente d\303\251finie pas 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.plot([x,f(x),x=-3..4]);Le trac\303\251 du graphe de cette fonction ne nous donne pas n\303\251cessairement des informations claires quant \303\240 la continuit\303\251 de celle-ci. Il semble que cette fonction ne poss\303\250de aucune discontinuit\303\251. Calculons f(3).f(3);Puisque 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 n'est pas d\303\251finie, l'\303\251galit\303\251 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 ne peut donc \303\252tre v\303\251rifi\303\251e. La fonction f est donc discontinue en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjNGJ0Y+LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZYRmVuRj4=. Rappelons que 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 existe:Limit((x^3-4*x^2+3*x)/(x-3),x=3)=limit((x^3-4*x^2+3*x)/(x-3),x=3);La fonction f poss\303\250de donc un discontinuit\303\251 non essentielle en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjNGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+. Puisque f(3) n'est pas d\303\251finie, la fonction f pr\303\251sente un trou, on parle alors d'une discontinuit\303\251 par trou.Si on d\303\251finissait f(3) de telle mani\303\250re que 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, cette discontinuit\303\251 non essentielle serait qualifi\303\251e de discontinuit\303\251 par d\303\251placement.discont(f(x),x);La macro-commande Maple discont renvoie tous les points de discontinuit\303\251 sur les r\303\251els. Cela inclut les points o\303\271 la fonction va \303\240 plus ou moins l'infini. Donc, ce r\303\251sultat ne nous informe pas sur la nature de la discontinuit\303\251: essentielle ou non essentielle.Dans le cas d'une discontinuit\303\251 par trou, il est possible de prolonger la d\303\251finition de la fonction f en donnant \303\240 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 la valeur de 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 pour en faire une fonction continue en 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. Ainsi la fonction f nouvellement d\303\251finie 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 continue en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjNGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+. Nous venons de prolonger f par continuit\303\251 en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKSZlcXVhbHM7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIzRidGOS8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGPUY5.Rigoureusement, en proc\303\251dant ainsi, nous faisons un abus de notation. En effet, la fonction f, que nous prolongeons par continuit\303\251 en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjNGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+, n\342\200\231est pas d\303\251finie en ce r\303\251el. Lorsque nous la prolongeons par continuit\303\251, le domaine de d\303\251finition de la fonction change et donc la fonction (math\303\251matiquement parlant) change. Nous devrions changer de notation en toute rigueur.En pratique, il y a tr\303\250s souvent int\303\251r\303\252t \303\240 prolonger (lorsque cela est possible) une fonction par continuit\303\251 car la fonction obtenue (\303\251tant continue) va satisfaire des th\303\251or\303\250mes et propri\303\251t\303\251s facilitant l\342\200\231\303\251tude math\303\251matique.Le graphique de la fonction 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 = 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 ressemble beaucoup au graphique d'une fonction quadratique, c'est-\303\240-dire \303\240 une parabole. En effet, observons le calcul de simplification suivant.f(x);simplify(f(x));On constate donc que le calcul de 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 est \303\251gale au calcul de 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 qu'\303\240 la condition \303\251videmment que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJm5lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiM0YnRj4vJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRkJGPkYrRlhGZW5GPg==. Exercice Maple, v\303\251rifions cela en posant, dans les deux expressions pr\303\251c\303\251dentes, 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 .'f(sqrt(2))'=f(sqrt(2));Rationnalisons 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.``= radnormal(rhs(),'rationalized');
Et Eval((x-1)*x,x=sqrt(2))=eval((x-1)*x,x=sqrt(2));
expand(rhs(%));Tra\303\247ons maintenant le graphique de la fonction "en exag\303\251rant" la discontinuit\303\251 r\303\251ductible en x=3.Trou:=plottools[disk]([3,6],0.15,color=white):
Courbe:=plot([x,f(x),x=-3..4]):
plots[display]([Courbe,Trou],scaling=constrained);Remarque: Lors du trac\303\251 d'un graphique, Maple calcule les coordonn\303\251es d'un certain nombre de points selon un algorithme. Rappelons qu'ensuite, il relie ces points selon son m\303\251canisme d'interpolation. Alors, pour des graphiques avec discontinuit\303\251s r\303\251ductibles, c'est seulement par analyse qu'il est possible de trouver les valeurs de discontinuit\303\251 de la fonction.Autres exemples de discontinuit\303\251 essentielles (par saut, infinie et par oscillations)Discontinuit\303\251 par sautVoici un premier exemple d'une fonction discontinue par saut. Tra\303\247ons la fonction partie enti\303\250re de x: 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 .f:=x->floor(x);plot([x,f(x),x=-3..5]);Ce trac\303\251 n'est pas correct. Maple a trac\303\251 des droites presque verticales autour des points de discontinuit\303\251 (c'est d\303\273 au m\303\251canisme d'interpolation). L'option discont=true indiquera \303\240 Maple de faire attention \303\240 ce type de discontinuit\303\251 et ainsi de tracer correctement le graphique.plot([x,f(x),x=-3..5],discont=true);Cette fonction, dont le domaine est l'ensemble de tous les nombres r\303\251els, pr\303\251sente une infinit\303\251 de points de discontinuit\303\251: il s'agit de tous les points de la fonction pour lesquels l'abscisse est un nombre entier. C'est la deuxi\303\250me condition de la continuit\303\251 qui n'est pas satisfaite.La fonction partie enti\303\250re fournit donc un exemple de fonction d\303\251finie sur \342\204\235 et discontinue pour 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 (et donc pas continue sur \342\204\235).Voici un autre exemple d'une fonction discontinue par saut.f:=x->x-floor(x);plot([x,f(x),x=-1..3],view=[-1..3,0..1.5],discont=true);Voici un dernier exemple obtenu \303\240 l'aide de la fonction piecewise. ( fonction d\303\251finie par morceaux ).f:=t->piecewise(t<1,t^2,t>=1 and t<2,1,3);
'f'(t)=simplify(f(t));Remarque: il est int\303\251ressant de constater la mani\303\250re dont Maple a simplifi\303\251 l'\303\251nonc\303\251 de cette fonction. L'ordre des \303\251nonc\303\251s conditionnels sur les valeurs de la variable est TR\303\210S important. \303\200 la premi\303\250re condition vraie, Maple retourne imm\303\251diatement une valeur.Tra\303\247ons le graphique de f.plot([t,f(t),t=-3..4],discont=true);Cette fonction est continue partout sauf en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjJGJ0Y+LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZYRmVuRj4=. Car...f(2);limit(f(t),t=2);c'est la deuxi\303\250me condition qui n'est pas satisfaite.En effet, la limite n'existe pas car la limite \303\240 gauche n'est pas \303\251gale \303\240 la limite \303\240 droite.Limit('f'(t),t=2,left)=limit(f(t),t=2,left);Limit('f'(t),t=2,right)=limit(f(t),t=2,right);Discontinuit\303\251 infinieConsid\303\251rons la fonction LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIzRidGPi1JJW1zdXBHRiQ2Ji1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRInhGJ0Y0RjctRjs2LVEnJnBsdXM7RidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjIyMjIyMjJlbUYnL0ZTRmRvLUZYNiRRIjJGJ0Y+LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GPkZmby8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSJeRicvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmlwLyUpYmV2ZWxsZWRHRkJGaW9GXHBGPkYrRmlvRlxwRj4= .g:=x->3/(x+2)^2:
plot([x,g(x),x=-3..1]);On a que dom f = \342\204\235\342\210\226LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtb0dGJDYtUSgmbWludXM7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkgtSSNtbkdGJDYkUSIyRidGNEY0RjQvJSVvcGVuR1EnJmxjdWI7RicvJSZjbG9zZUdRJyZyY3ViO0YnLyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y5RjQ= puisque 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 n'est pas d\303\251finie, la fonction est donc discontinue en 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. La premi\303\250re condition de la continuit\303\251 n'est pas satisfaite.On observe, par ailleurs, que pour les valeurs de x de plus en plus voisines de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjJGJ0Y1LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y6RjVGK0ZQRlNGNQ==, tout en \303\251tant inf\303\251rieures \303\240 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjJGJ0Y1LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y6RjVGK0ZQRlNGNQ==, lorsque 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEtJnJpZ2h0YXJyb3c7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkQtSSNtbkdGJDYkUSIwRidGLy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJ0Yv , leurs images ont des valeurs de plus en plus grandes: ces images ne deviennent pas de plus en plus voisines d'un certain nombre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZpc2ludjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSgmIzg0Nzc7RicvRjBGPUY5LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y9Rjk=. Donc la limite \303\240 gauche n'existe pas. Dans ce cas, m\303\252me si la limite \303\240 gauche n'existe pas, il est permis d'utiliser cette notation de limite pour donner une information quant au comportement des images de ces valeurs de x voisines de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjJGJ0Y1LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y6RjVGK0ZQRlNGNQ==:Limit('g'(x),x = -2,left)=limit(g(x),x = -2,left);Il faut donc comprendre cet \303\251nonc\303\251 comme ceci: lorsque les valeurs de x sont de plus en plus voisines de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjJGJ0Y1LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y6RjVGK0ZQRlNGNQ==, tout en \303\251tant inf\303\251rieures \303\240 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjJGJ0Y1LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y6RjVGK0ZQRlNGNQ==, lorsque LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtZmVuY2VkR0YkNigtRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZULUYwNiQtRiM2JUY9LUkjbW5HRiQ2JFEiMkYnRkFGQUZBRkFGQS9JK21zZW1hbnRpY3NHRiRRJGFic0YnLyUlb3BlbkdRKSZ2ZXJiYXI7RicvJSZjbG9zZUdGXm9GaW4vJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRkVGQQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEtJnJpZ2h0YXJyb3c7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkQtSSNtbkdGJDYkUSIwRidGLy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJ0Yv leurs images prennent des valeurs de plus en plus grandes.De mani\303\250re similaire, demandons \303\240 Maple de calculer la limite \303\240 droite:Limit('g'(x),x = -2,right)=limit(g(x),x = -2,right);En r\303\251sum\303\251, puisque le comportement des images des voisins de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjJGJ0Y1LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y6RjVGK0ZQRlNGNQ==, aussi bien les voisins de gauche que ceux de droite, est le m\303\252me, on peut \303\251crire:Limit('g'(x),x = -2)=limit(g(x),x = -2);Dans les manuels, on a l'habitude, dans de tels cas de discontinuit\303\251, de tracer \303\251galement une droite verticale qu'on appelle asymptote verticale pour mettre davantage l'emphase sur le comportement des images des valeurs voisines de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Jy1JI21vR0YkNi1RKCZtaW51cztGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjJGJ0Y1LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0Y6RjVGK0ZQRlNGNQ==. Ajoutons manuellememt le trac\303\251 de l'asymptote verticale d'\303\251quation 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. DroiteV:=plot([-2,y,y=-1..100],linestyle=3,color="Niagara Burgundy"):
Graphe:=plot([x,g(x),x=-3..0]):
plots[display]({DroiteV,Graphe},view=[-3..3,-1..100]);Dans le cas d'une discontinuit\303\251 infinie o\303\271 le comportement des images des voisins de gauche et de droite d'une valeur de x n'est pas le m\303\252me, on ne peut pas utiliser la notation limite sans pr\303\251ciser la "l'approche" des valeurs de x pour donner l'information quant au comportement des images de tous les voisins de x.Examinons la fonction suivante:f:=x->1/(x-1);plot([x,f(x),x=-1..3],discont=true,view=[-1..3,-20..20]);Limit('f'(x),x = 1,left)=limit(f(x),x = 1,left);Limit('f'(x),x = 1,right)=limit(f(x),x = 1,right);Essayons de voir ce que r\303\251pondra Maple lorsqu'on lui demandera d'\303\251valuer la limite sans pr\303\251ciser "l'approche" des voisins de 1. Pouvez-vous le deviner?Limit('f'(x),x = 1)=limit(f(x),x = 1);Maple a r\303\251pondu "undefined" mais nous savions d\303\251j\303\240 que limite n'existe pas car la limite \303\240 gauche n'existe pas. Une autre bonne fa\303\247on de r\303\251pondre aurait \303\251t\303\251 de dire que la limite n'existe pas car la limite \303\240 droite n'existe pas.Discontinuit\303\251 par oscillationsTerminons cette pr\303\251sentation en observant la fonction 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 .f:=x->sin(1/x):
plot([x,f(x),x=-Pi..Pi],view=[-3..3,-2..2]);Bien s\303\273r, 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 n'\303\251tant pas d\303\251finie, la premi\303\250re condition de la continuit\303\251 n'est pas satisfaite. Donc cette fonction n'est pas continue en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjBGJ0Y+LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZYRmVuRj4=. Mais analysons le type de discontinuit\303\251 de cette fonction en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjBGJ0Y+LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZYRmVuRj4=. Tra\303\247ons, de nouveau, le graphique de f en limitant davantage l'intervalle des valeurs de x.plot([x,f(x),x=-0.5..0.5],view=[-0.5..0.5,-1.5..1.5]);plot([x,f(x),x=-0.01..0.01],view=[-0.01..0.01,-1.5..1.5]);Il semble que les images des voisins de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjBGJ0Y+LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZYRmVuRj4= ne deviennent ni de plus en plus voisines d'un nombre L, ni de plus en plus grandes. On remarque plut\303\264t que ces images oscillent continuellement entre les droites 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 et LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZYRmVuRj4=. Interrogeons Maple pour voir. Int\303\251ressons-nous d'abord aux voisins de gauche de 0. Limit('f'(x),x=0,left)=limit(f(x),x=0,left);Maple r\303\251pond, \303\240 sa fa\303\247on, que la limite demand\303\251e n'existe pas. De plus, il nous informe que les images de ces valeurs de x voisines mais inf\303\251rieures \303\240 0 oscillent entre 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 et LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZCRj5GK0ZYRmVuRj4=. La r\303\251ponse donn\303\251e est la notation Maple pour un objet de type intervalle ferm\303\251.De la m\303\252me mani\303\250re, pour les voisins de droite,Limit('f'(x),x=0,right)=limit(f(x),x=0,right);En r\303\251sum\303\251:Limit('f'(x),x=0)=limit(f(x),x=0);En bref, le comportement des images des valeurs de x pr\303\250s de 0 est le m\303\252me aussi bien que avec les voisins de gauche que pour ceux de droite de 0.M\303\252me apr\303\250s quelques trac\303\251s de f en voulant zoomer vers l'avant, on ne voit pas grand chose sur le graphique pr\303\251c\303\251dent au voisinage de 0. Voici une mani\303\250re de mieux voir le comportement "erratique" des images en zoomant vers l'avant le graphique de f au voisinage de 0.On g\303\251n\303\250re d'abord une suite de valeurs de x voisines de 0. On contr\303\264le alors les ant\303\251c\303\251dents de la fonction f pour son trac\303\251 au voisinage de 0.Points:= evalf[40]([seq]([-0.01 + i/200*0.04,f(-0.01 + i/200*0.04)],i=0..200)):
plots[pointplot](Points,style=line,connect=true,view=[-0.01..0.01,-1.5..1.5],color="Niagara Burgundy"); Le mot de la finPresque toutes les fonctions de notre niveau d'enseignement sont continues sur tout intervalle contenu dans leur domaine de d\303\251finition :les fonctions polynomiales sont continues sur \342\204\235 ;les fonctions rationnelles sont continues sur tout intervalle contenu dans leur domaine de d\303\251finition ;les fonctions exponentielles sont continues sur \342\204\235 ;les fonctions logarithme n\303\251p\303\251rien et logarithme d\303\251cimal sont continues sur ]0, +\342\210\236[ ;les fonctions racines n-\303\250mes sont continues sur [0, +\342\210\236[ ;les fonctions sinus et cosinus sont continues sur \342\204\235, la fonction tangente est continue sur tout intervalle ne contenant pas un nombre de la forme 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, k \342\210\210 Z.la fonction valeur absolue est continue sur \342\204\235 ; toutes les fonctions obtenues par op\303\251rations (somme, produit, quotient) ou composition \303\240 partir de ces fonctions de r\303\251f\303\251rence sont aussi continues sur leur domaine de d\303\251finition.Portons maintenant notre attention sur la fonction f d\303\251finie par 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.Obtenons le domaine de la fonction f.
Pour que la racine carr\303\251e puisse \303\252tre d\303\251finie, il faut que le radicand repr\303\251sente un nombre positif. Alors, r\303\251solvons l'in\303\251quation 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. (La relation \342\211\245 impose une r\303\251solution dans \342\204\235)solve(x^2+x-6>=0,x);La fonction f est continue sur son domaine donc la fonction f est continue sur ]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] \342\213\203 [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSgmY29tbWE7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSgmaW5maW47RicvJSdpdGFsaWNHRj5GNUY1RisvJSVzaXplR1EjMTJGJy8lK2V4ZWN1dGFibGVHRj5GNQ==[. \303\211videmment, la continuit\303\251 en 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 est une continuit\303\251 \303\240 gauche et la continuit\303\251 en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEpJmVxdWFscztGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjJGJ0Y+Rj5GKy8lJXNpemVHUSMxMkYnLyUrZXhlY3V0YWJsZUdGQkY+ est une continuit\303\251 \303\240 droite. f:=x->:-sqrt(x^2+x-6);Limit(f(x),x=-3,left)=limit(f(x),x=-3,left);
Limit(f(x),x=2,right)=limit(f(x),x=2,right);plot([x,f(x),x=-6..5],numpoints=500,xtickmarks=10,view=[-6..5,0..5]);Observons le r\303\251sultat, ci-dessous, \303\240 la requ\303\252te suivante: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.Limit(f(x),x=-3,right)=limit(f(x),x=-3,right);\303\200 l'aide de LimitTutor de la sous-biblioth\303\250que Student[Calculus1], voyons comment comment l'\303\251valuateur a op\303\251r\303\251 ce calcul. \303\200 l'ex\303\251cution de LimitTutor appara\303\256tra une bo\303\256te de dialogue. Cliquez sur le bouton Toutes les \303\251tapes puis ensuite, sur le bouton Fermer. \303\200 la fermeture de la bo\303\256te de dialogue, tout le d\303\251veloppement sera transpos\303\251 dans le document appelant.LimitTutor((f(x),x=-3,right));Le domaine de simplification de Maple est l'ensemble des nombres complex mais, selon la documentation, la sous-biblioth\303\250que Student[Calculus1] devrait r\303\251aliser les simplifications dans l'ensemble des nombres r\303\251els. Contrairement \303\240 la documentation, il semble que Maple ait r\303\251alis\303\251 tout de m\303\252me un passage \303\240 la limite dans \342\204\202. Ce r\303\251sultat n'est pas celui attendu dans \342\204\235. Le passage \303\240 la limite qu'a op\303\251r\303\251 l'\303\251valuateur dans le calcul pr\303\251c\303\251dent n'est pas acceptable dans \342\204\235. En effet, si 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGWC1GOzYkUSI5RidGPi1GRDYtUSI8RidGPkZHRkpGTEZORlBGUkZUL0ZXUSwwLjI3Nzc3NzhlbUYnL0ZaRlxvLUY7NiRGQkY+LyUlc2l6ZUdRIzEyRicvJStleGVjdXRhYmxlR0ZJRj4= et donc, 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.Limit(f(x),x=-3,right)=limit(f(x),x=-3,right);
Limit(f(x),x=2,left)=limit(f(x),x=2,left);Avec l'environnement RealDomain, on a ceci:use RealDomain in Limit(f(x),x=-3,right)=limit(f(x),x=-3,right) end use;
use RealDomain in Limit(f(x),x=-3,left)=limit(f(x),x=-3,left) end use;Mais, ceci aussi.use RealDomain in Limit(f(x),x=2,left)=limit(f(x),x=2,left) end use;
use RealDomain in Limit(f(x),x=2,right)=limit(f(x),x=2,right) end use;La le\303\247on qu'on doit retenir c'est qu'il est primordial de consid\303\251rer le domaine d'une fonction avant d'en faire l'analyse. La macro-commande discont a pour r\303\251sultat non pas des discontinuit\303\251s av\303\251r\303\251es mais des candidats \303\240 la discontinuit\303\251.discont(f(x),x);Or, la fonction f est discontinue sur 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.