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h + o(h), \qquad f_2(\xbar + h) = f_2(\xbar) + f_2'(\xbar) h + o(h). \] \begin{enumerate}[(i)] @@ -156,12 +153,11 @@ \begin{enumerate}[(i)] \item $\ybar$ è un punto di accumulazione di $Y$, - \item $g$ è derivabile in $\ybar$, - \item $g'(\ybar) = \frac{1}{f'(\xbar)}$. + \item $g$ è derivabile in $\ybar$ e $g'(\ybar) = \frac{1}{f'(\xbar)}$. \end{enumerate} \end{proposition} - \begin{proof}\nl + \begin{proof} Si dimostrano i due risultati separatamente. \begin{enumerate}[(i)] \item Poichè $f$ è derivabile in $\xbar$, $f$ è continua in $\xbar$. Quindi per ogni intorno $I$ di $\ybar$, esiste @@ -169,7 +165,7 @@ è mai vuoto, dacché, essendo $f$ derivabile in $\xbar$, $\xbar$ è un punto di accumulazione di $X$. Quindi $J$ contiene in particolare un immagine di $f$ in esso, e quindi un punto di $Y$; inoltre, tale punto è diverso da $\ybar$ dal momento che $f$ è iniettiva, essendo bigettiva. Quindi $\ybar$ è un punto di accumulazione. - \item e \!(iii) Poiché $f$ è derivabile in $g(\ybar)$, + \item Poiché $f$ è derivabile in $g(\ybar)$, $\ybar + h = f(g(\ybar + h)) = f(g(\ybar) + (\underbrace{g(\ybar + h) - g(\ybar)}_k)) = \ybar + f'(\xbar) k + o(k)$, ossia vale che: @@ -188,6 +184,8 @@ \[ f(x) = \system{x & \se x \geq 0, \\ -(x+2) & \se -2 < x \leq -1.} \] + \vskip 0.05in + dove $f'(0) = 1$, $f$ è invertibile, ma la derivata di $g$ in $0$ non esiste ($D_+ g(0) = 1$, ma $D_- g(0) = +\infty$). \end{example} @@ -198,10 +196,34 @@ $f'(\xbar) = 0$. \end{theorem} - \begin{example} - Dimostrare che la derivata sinistra è negativa, e che quella - destra è positiva nei casi che hai capito. - \end{example} + \begin{proof} + Poiché $I$ è un intervallo e $\xbar$ è interno a $I$, $\xbar$ è sia punto di accumulazione + sinistro che punto di accumulazione destro di $I$. Dal momento che $f$ è derivabile + in $\xbar$, esistono sia la derivata destra che la derivata sinistra in $\xbar$. \\ + + Si assuma che $\xbar$ è un punto di massimo locale (altrimenti è sufficiente considerare $g = -f$). + Allora esiste un intorno $J$ di $\xbar$ tale per cui $x \in J \implies f(x) - f(\xbar) \leq 0$. Sia dunque + $J_+$ l'intorno destro relativo a $J$, e sia $J_-$ quello sinistro. \\ + + Poiché $\xbar = \inf J_+$, esiste una successione $\{x_n\} \subseteq J_+ \setminus \{\xbar\}$ tale + per cui $x_n \tendston \xbar$. Dal momento che allora $f$ è derivabile in $\xbar$, $f$ è anche + continua in $\xbar$, e quindi si ricava che $f(x_n) \tendston f(\xbar)$. Si osserva dunque + che $f(x_n) - f(\xbar) \leq 0$ e $x_n - \xbar > 0 \implies \frac{f(x_n) - f(\xbar)}{x_n - \xbar} \leq 0$, + da cui, per il teorema della permanenza del segno, si ricava che $L_+ = \lim_{n \to \infty} \frac{f(x_n) - f(\xbar)}{x_n - \xbar} \leq 0$. \\ + + Allora, dal momento che $f$ è derivabile in $\xbar$ e che la derivata destra + deve coincidere con la derivata classica, $f'(\xbar) = + \lim_{x \to \xbar^+} \frac{f(x) - f(\xbar)}{x-\xbar} = \lim_{n \to \infty} \frac{f(x_n) - f(\xbar)}{x-\xbar} = L_+ \leq 0$. \\ + + Analogamente si ricava che $f'(\xbar) \geq 0$, e quindi che $f'(\xbar)$ è necessariamente pari a zero, da cui la + tesi. + \end{proof} + + \begin{remark} \nl + \li Si può facilmente generalizzare il teorema di Fermat assumendo ipotesi più deboli. Sia infatti $x_M$ un punto di massimo locale e sia $f$ continua in $x_M$, + allora, qualora esistano, $D_+ f(x_M) \leq 0$ e $D_- f(x_M) \geq 0$. Analogamente + si estende la proposizione a $x_m$ punto di minimo locale. + \end{remark} \begin{theorem} (di Rolle) Sia $I = [a, b] \subset \RR$ e sia $f : I \to \RR$ tale che @@ -212,11 +234,10 @@ \begin{proof} Per il teorema di Weierstrass $f$ ammette un punto di massimo $M$ e uno di minimo $m$ in $I$. Se $f(a) = M$ e $f(b) = m$ o viceversa, la funzione $f$ è costante in $I$, e quindi per ogni punto in $(a, b)$ - la derivata è nulla, dacché $f$ è sempre derivabile. Altrimenti, + la derivata è nulla. Altrimenti, sicuramente uno tra il punto di massimo e quello di minimo appartiene - a $(a, b)$. Senza perdita di generalità, si assuma che $\exists x_M \in (a, b)$ tale che $f(x_M) = M$: per - il teorema di Fermat $f'(x_M) = 0$. Analogamente per il caso in cui - $\exists x_m \in (a, b)$ tale che $f(x_m) = m$, da cui la tesi. + a $(a, b)$. Sia $\xbar$ tale punto. Allora, per il teorema di Fermat, $f'(\xbar) = 0$, da cui + la tesi. \end{proof} \begin{theorem} (di Cauchy) @@ -238,12 +259,12 @@ \begin{theorem} (di Lagrange) Sia $I = [a, b] \subset \RR$ e sia $f: I \to \RR$ tale che $f$ sia continua su $I$ e che $f$ sia derivabile in $(a, b)$. Allora - $\exists \xbar \in (a, b)$ tale che $f'(\xbar) = \frac{f(b) - f(a)}{b-a}$, ossia la cui retta tangente è parallela alla secante + $\exists \xbar \in (a, b)$ tale che $f'(\xbar) = \frac{f(b) - f(a)}{b-a}$, ossia tale per cui la retta tangente a $f$ in $\xbar$ è parallela alla secante che passa per $(a, f(a))$ e $(b, f(b))$. \end{theorem} \begin{proof} - Si consideri $g(x) = x$, $g$ è continua in $[a, b]$ e derivabile + Si consideri $g(x) = x$. $g$ è continua in $[a, b]$ e derivabile in $(a, b)$, con derivata sempre non nulla in tale intervallo. Allora, per il teorema di Cauchy, $\exists \xbar \in (a, b) \mid f'(\xbar) = \frac{f(b)-f(a)}{b-a}$, da cui la tesi. @@ -263,13 +284,13 @@ Si considerino $c < d \in I$. Allora, per il teorema di Lagrange, $\exists \xbar \in (c, d) \mid f'(c) = \frac{f(d) - f(c)}{d-c} \implies f(d) - f(c) = \underbrace{f'(c) (d-c)}_{\geq 0} \implies - f(d) \geq f(c)$, ossia che $f$ è crescente in $I$. + f(d) \geq f(c)$, e quindi $f$ è crescente in $I$, da cui la tesi. \end{proof} \begin{remark}\nl \li L'interpretazione geometrica del teorema di Cauchy, rispetto a quella di Lagrange, è leggermente più complicata. Si consideri - la curva $\gamma : \RR \to \RR^2$ tale che + la curva continua $\gamma : \RR \to \RR^2$ tale che $\gamma(t) =(g(t), f(t))$. Si osserva che il coefficiente della retta tangente in $\xbar$ per $\gamma$ è dato da $\lim_{h \to 0} \frac{f(\xbar + h) - f(\xbar)}{g(\xbar + h) - g(\xbar)}$, che, sotto le ipotesi del teorema di Cauchy, può essere riscritto @@ -280,43 +301,68 @@ \end{remark} \begin{exercise} - Dare un esempio di una funzione $f : \RR \to \RR$ crescente e - discontinua $\forall x \in \ZZ$. + Si descriva un insieme $X$ tale che i suoi unici punti di accumulazione + siano $\pm 1$. \end{exercise} \begin{solution} - Si consideri $f(x) = \lfloor x \rfloor$. + Si consideri $X = \{1 + \frac{1}{n}\} \cup \{-1 - \frac{1}{n}\}$, al variare di $n \in \NN$. Sia + $J = [1 - \eps, 1 + \eps]$ un intorno di $1$. Allora $1 + \frac{1}{n} \in J$ per $n > \frac{1}{\eps}$, + da cui si ricava che $1$ è un punto di accumulazione di $X$; analogamente si verifica che $-1$ è + un punto di accumulazione di $X$. Si consideri adesso l'intorno $J = \left[1 + \frac{1}{n} - \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+1} \right), 1 + \frac{1}{n} + \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+1} \right) \right]$. Si verifica che nessun punto di $X$, oltre $1 + \frac{1}{n}$ appartiene a $J$, e quindi + $1 + \frac{1}{n}$ non è punto di accumulazione di $X$. Analogamente non lo è alcun numero della forma + $-1 - \frac{1}{n}$. \end{solution} \begin{exercise} - Si descriva un insieme $X$ tale che i suoi punti di accumulazione - sono $\{\pm 1\}$. + Sia $f : X \to \RRbar$ continua in $\xbar$ e sia $a < f(\xbar)$. + Allora esiste $J$ intorno di $\xbar$ tale che $a < f(x)$ $\forall + x \in J \cap X \setminus \{\xbar\}$. \end{exercise} \begin{solution} - Si consideri $X = \{1 + \frac{1}{n}\} \cup \{-1 + \frac{1}{n}\}$. + Si consideri $g : X \to \RRbar$ tale che $g(x) = f(x) - a$. Poiché $g$ è una somma di + funzioni continue in $\xbar$, anch'essa è continua in $g$. Allora, poiché $g(\xbar) > 0$, + per il teorema della permanenza del segno, esiste un intorno $J$ di $\xbar$ tale per cui + $g(x) > 0$ $\forall x \in J$, ossia tale per cui $f(x) > a$ $\forall x \in J$, da cui la tesi. \end{solution} \begin{exercise} - Sia $f : X \to \RRbar$ continua in $\xbar$ e sia $a < f(\xbar)$. - Allora esiste $J$ intorno di $\xbar$ tale che $a < f(x)$ $\forall - x \in J$. + Sia $X \subseteq \RRbar$ e sia $\xbar$ punto di accumulazione di $X$. Siano + $f_1$, $f_2 : X \to \RRbar$. Si dimostri allora che: + + \begin{enumerate}[(i)] + \item se $f_1 \tendsto{\xbar} +\infty$ e $f_2$ è limitata inferiormente + in un intorno $J$ di $\xbar$, allora $f_1(x) + f_2(x) \tendsto{\xbar} +\infty$; + \item se $f_1 \tendsto{\xbar} 0$ e $f_2$ è limitata in un intorno $J$ + di $\xbar$, allora $f_1 f_2(x) \tendsto{\xbar} 0$; + \item se $f_1 \tendsto{\xbar} +\infty$ è limitata inferiormente + da una costante positiva $c$ in un intorno $J$ di $\xbar$, allora + $f_1 f_2 \tendsto{\xbar} +\infty$. + \end{enumerate} \end{exercise} - \begin{exercise} - Sia $X \subseteq \RRbar$ e sia $\xbar$ punto di accumulazione di $X$, - $f_1$, $f_2 : X \to \RRbar$. Allora: + \begin{solution} Si dimostrano i tre risultati separatamente. - \begin{enumerate}[(i)] - \item Se $f_1 \tendsto{\xbar} +\infty$ e $f_2$ è limitata inferiormente - in un intorno $J$ di $\xbar$, allora $f_1(x) + f_2(x) \tendsto{\xbar} +\infty$. - \item Se $f_1 \tendsto{\xbar} 0$ e $f_2$ è limitata in un intorno - di $\xbar$, allora $f_1 f_2(x) \tendsto{\xbar} 0$. - \item Se $f_1 \tendsto{\xbar} +\infty$ è limitata inferiormente - da una costante positiva $m$ in un intorno $J$ di $\xbar$, allora - $f_1 f_2 \tendsto{\xbar} +\infty$. - \end{enumerate} - \end{exercise} + \begin{enumerate}[(i)] + \item Sia $c$ la costante tale per cui $f_2(x) \geq c$ $\forall x \in J \cap X$. Sia $I = [a, \infty]$ un intorno di $+\infty$. Se $c < 0$, poiché $f_1 \tendsto{\xbar} +\infty$, esiste un intorno $J'$ tale per cui $f_1(J' \cap X \setminus \{ \xbar \}) \subseteq [a-c, \infty] \subseteq I$. Sia dunque $Z = J \cap J'$. Allora + $(f_1 + f_2)(x) = f_1(x) + f_2(x) \geq a-c+c = a$ $\forall x \in Z$, da cui si conclude che + $(f_1 + f_2)(Z \cap X \setminus \{\xbar\}) \subseteq I$. Se invece $c \geq 0$, è sufficiente considerare + un intorno $J'$ di $\xbar$ tale per cui $f_1(J' \cap X \setminus \{\xbar\}) \subseteq I$, da cui + $(f_1 + f_2)(x) = f_1(x) + f_2(x) \geq a+c \geq a$ $\forall x \in Z \implies (f_1+f_2)(Z \cap X \setminus \{\xbar\}) \subseteq I$, da cui la tesi. + + \item Poiché $f_2$ è limitata in $J$, esistono delle costanti finite $a$, $b \in \RR$ tali per cui $a \leq f_2(x) \leq b$ $\forall x \in J$. Sia + $I = [-\eps, \eps]$ un intorno di $0$, con $\eps > 0$. Si consideri $c := \max\{\abs a, \abs b\}$. + Allora vale che $-c \leq f_2(x) \leq c$ $\forall x \in J$. Poiché $f_1 \tendsto{\xbar} 0$, esiste + un intorno $J'$ di $\xbar$ tale per cui $f(J' \cap X \setminus \{\xbar\}) \subseteq \left[-\frac{\eps}{c}, \frac{\eps}{c}\right]$. Si consideri ora $Z := J \cap J'$: vale allora che $\abs{(f_1 f_2)(x)} = \abs{f_1(x) f_2(x)} \leq c \frac{\eps}{c} = \eps$ $\forall x \in Z \cap X \setminus \{\xbar\}$. Si conclude + dunque che $(f_1f_2)(Z \cap X \setminus \{\xbar\}) \subseteq I$, da cui la tesi. + + \item Sia $I = [a, \infty]$ un intorno di $+\infty$. Allora, poiché $f_1 \tendsto{\xbar} +\infty$, + esiste un intorno $J'$ di $\xbar$ tale per cui $f_1(J' \cap X \setminus \{\xbar\}) \subseteq [\abs a, \infty] \subseteq I$. Si + consideri dunque $Z := J \cap J'$: vale dunque che $(f_1f_2)(x) = f_1(x) f_2(x) \geq \abs{a}c \geq a$ $\forall x \in Z \cap X \setminus \{\xbar\}$. Si conclude allora che $(f_1f_2)(Z \cap X \setminus \{\xbar\}) \subseteq I$, da cui la tesi. + \end{enumerate} + + \end{solution} \begin{exercise} Sia $f: \RR \to \RR$ tale che: @@ -325,6 +371,26 @@ \vskip 0.05in - Mostrare che $f$ è continua, che $f'(0) = 1$ e che $f'$ non è continua in zero. + Si mostri che $f$ è continua ovunque e che $D_+ f(0) = 1$. \end{exercise} + + \begin{solution} + Poiché somma di funzioni elementari, $f$ è continua in $(0, \infty)$. Analogamente è continua in + $(-\infty, 0)$ dacché è costante in tale intervallo. Affinché allora $f$ sia continua ovunque è + sufficiente che si dimostri che è continua anche in $0$. Dal momento che $0$ è un punto di accumulazione + sia destro che sinistro di $\RR$, questo equivale a mostrare che il limite destro e sinistro di $f$ + esistono in $0$ e coincidono. \\ + + Si verifica dunque che + \[ \lim_{x \to 0^-} f(x) = 0, \qquad \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x + \lim_{x \to 0^+} 2x^2 \sin\left(\frac{1}{x}\right) = 0, \] + + dove si è impiegato il fatto che $\sin\left(\frac{1}{x}\right)$ è limitata in ogni intorno di $0$ e che + $2x^2 \tendsto{0^+} 0$; quindi $f$ è continua in $0$, e lo è allora ovunque. \\ + + Si computa allora la derivata destra di $f$ in $0$: + + \[ D_+ f(0) = \lim_{h \to 0^+} \frac{h + 2h^2 \sin\left(\frac{1}{h}\right)}{h} = 1 + \lim_{h \to 0^+} 2 h \sin\left(\frac{1}{h}\right) = 1, \] + + dove si è usato lo stesso argomento di prima per computare $\lim_{h \to 0^+} 2 h \sin\left(\frac{1}{h}\right) = 0$. \\ + \end{solution} \end{document} \ No newline at end of file