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zoF4%CO9;ZY|Fz^Fr4ep|Jyr&hC4aR;2ng|{K>uI;`hQalfQTc$|ER=Gyd2>fdt5Xkqa zAR!)k0DlRFBIG`HCmhjvKz|8=0RFAvKMF$se&&I~{%M&1W)RT;K>usq|ICFUa^f!x z!52dAe?Bk4fQZMpf2jFqGNSlDrVz%DC^e7AAv)A!7XB5}k7EJRe*X`HDKZ}sXN7@4 zAPGrHQ87tG-iyN^{JdasAOwi0JtAUYQN$-9`j`LTA-eQJLfDQ@h7L|{4yI<<02l<( NWU!f;B@`sF{~xpW!z};+ diff --git a/Analisi matematica 1/Parte teorica/2023-03-23, 24, 28, Proprietà principali della continuità e dei limiti di funzione/main.tex b/Analisi matematica 1/Parte teorica/2023-03-23, 24, 28, Proprietà principali della continuità e dei limiti di funzione/main.tex index cad1b44..8b6545d 100644 --- a/Analisi matematica 1/Parte teorica/2023-03-23, 24, 28, Proprietà principali della continuità e dei limiti di funzione/main.tex +++ b/Analisi matematica 1/Parte teorica/2023-03-23, 24, 28, Proprietà principali della continuità e dei limiti di funzione/main.tex @@ -352,9 +352,9 @@ è però per forza vero il contrario, è sufficiente considerare $(0, \infty)$, dove $0$ è solo un punto di accumulazione destro), \\ \li $f$ è continua in $\xbar$ $\iff$ $f$ è continua sinistra e destra in $\xbar$, \\ - \li se $\xbar$ è un punto di accumulazione destro e sinistro, $\lim_{x \to \xbar} f(x) = L \iff \lim_{x \to \xbar^+} f(x) = L$ e $\lim_{x \to \xbar^-} f(x) = L$, \\ - \li se $\xbar$ è un punto di accumulazione solo destro, $\lim_{x \to \xbar} f(x) = L \iff \lim_{x \to \xbar^+} f(x) = L$, \\ - \li se $\xbar$ è un punto di accumulazione solo sinistro, $\lim_{x \to \xbar} f(x) = L \iff \lim_{x \to \xbar^-} f(x) = L$. + \li Se $\xbar$ è un punto di accumulazione destro e sinistro, $\lim_{x \to \xbar} f(x) = L \iff \lim_{x \to \xbar^+} f(x) = L$ e $\lim_{x \to \xbar^-} f(x) = L$, \\ + \li Se $\xbar$ è un punto di accumulazione solo destro, $\lim_{x \to \xbar} f(x) = L \iff \lim_{x \to \xbar^+} f(x) = L$, \\ + \li Se $\xbar$ è un punto di accumulazione solo sinistro, $\lim_{x \to \xbar} f(x) = L \iff \lim_{x \to \xbar^-} f(x) = L$. \end{remark} \begin{proposition} @@ -403,20 +403,25 @@ \end{example} \begin{exercise} - Mostrare che l'insieme dei punti di discontinuità di una funzione $f : I \to \RR$ monotona è al più + Mostrare\footnote{La tesi altro non è che un caso particolare del cosiddetto \textit{teorema di Froda}.} che l'insieme dei punti di discontinuità di una funzione $f : I \to \RR$ monotona è al più numerabile, dove $I$ è un intervallo. \end{exercise} \begin{solution} Si assuma $f$ crescente, senza perdita di generalità (altrimenti è sufficiente considerare $g = -f$). Sia $E$ l'insieme dei punti di discontinuità di $f$. $\forall \xbar \in E$, $\xbar$ è un punto di accumulazione - destro e sinistro di $I$ (infatti $I$ è un intervallo), ed in particolare esistono sempre il limite destro $L^-(\xbar)$ - ed il limite sinistro $L^+(\xbar)$ in $\xbar$ (dal momento che $f$ è monotona), e sono tali che $L^+(\xbar) > L^-(\xbar)$ (sicuramente - sono diversi, altrimenti + destro e sinistro di $I$ (infatti $I$ è un intervallo), ed in particolare, per la proposizione precedente, esistono sempre il limite sinistro $L^-(\xbar)$ + ed il limite destro $L^+(\xbar)$ in $\xbar$ (dal momento che $f$ è monotona), e sono tali che\footnote{Detti + $A = \{f(x) \mid x < \xbar \E x \in X\}$ e $B = \{f(x) \mid x > \xbar \E x \in X\}$, vale che $L^-(\xbar) = \sup A$ e $L^+(\xbar) = \inf B$. Dal momento che $f$ è crescente, vale che $B \geq A$. Se $\inf B < \sup A$, esisterebbe un $b \in B$ tale che $\sup A > b$, da cui ancora + esisterebbe un $a \in A$ tale che $a > b$, \Lightning. Quindi + $\inf B = L^+(\xbar) \geq \sup A = L^-(\xbar)$.} $L^+(\xbar) > L^-(\xbar)$ (infatti + sono sicuramente diversi, altrimenti $f$ sarebbe continua in $\xbar$; inoltre $f$ è crescente). Allora sia $f : E \to \QQ$ tale che $\xbar \mapsto c$, dove $c \in \QQ$ è un punto razionale in $(L^-(\xbar), L^+(\xbar))$ (tale $c$ esiste sempre, per la densità di $\QQ$ in $\RR$). - Inoltre, $x < y \implies L^+(x) \leq L^-(y)$, e quindi ogni intervallo da cui è preso $c$ è distinto al variare - di $\xbar \in E$. Quindi $f$ è iniettiva, e vale $\abs{E} \leq \abs{\QQ} = \abs{\NN}$. Si conclude allora + Inoltre\footnote{Sia $C = \{f(a) \mid x < a < y \}$. Allora $L^+(x) = \inf C$ e $L^-(y) = \sup C$. + Dal momento che $\sup C \geq \inf C$, deve allora valere anche + che $L^+(x) \leq L^-(y)$.}, $x < y \implies L^+(x) \leq L^-(y)$, e quindi ogni intervallo da cui $c$ è estratto è distinto al variare + di $\xbar \in E$. Pertanto $f$ è iniettiva, e vale che $\abs{E} \leq \abs{\QQ} = \abs{\NN}$. Si conclude allora che $E$ è al più numerabile. \end{solution}