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zzXeH1{2x~sxW>Q%kM9cN`U?>YAlla`hyM10MIeB~ z`~r|+76R}r@c0|2qUPqy-~{BLT8g zzyFebhy^v`V`n1(ARw6n4~7+R08GgV1aV>qGeC7UnTa4`0_ zz)86f2$)nz+s540+J#LRfi({S?D~s@CKaSZfDKCc8<8Fljun0ZXaZboCBQBe0GwK1 z0p^}017Ci<0$ssQ#{k+fMgA?azUF5RNE;9dL>X(55BLmx1j~^E=-^fYiU0u$ z5mbVB09&yc0L;7+Yumu_8?% z4h#$;jB^!=1|!hme^WrwXs|F)cmL!9ML{8G;PBOd!_Y_w1Q&)tL6QH4!N5>B__tgP zK>c5}plCP}iGv752_wclK|>L699kF{20=johxc1P@Dc{Y zaEQTRBpQZe3HUcL4jy0$;G6$3{LKxDh9HnQcu+J9f$Ie*3WY(!{-MoZ#3(offxuyi zLJA{sl!2lVC@_w7UYcT*r{ArvpkO9ACp#JB6Fa+=x>feF^7)%()I)EVxj=&za zfY>n*AQU);0wlVod7y`o;45$LdbtVv4BpS!Tzk`j!$!|az82o<*WBw{9C>jMu;p8I>Fa(Jk9hfi# zj%yv@Uf3UafLn2ufkRNZGQe;QMEH-0!GvLdSONzQHO3VTM<8$u3-Bpe7&jzvGz9qv zEzlxt5{~3&e!eAKO2%?|}G;Tuy<_PZ40HYy5UEwH$MhQc3vjk|g zxGfX{0aErqaxwpPKLCXNUL5~v9e@Y^x4Xb#00u?i2!Z@==r~~lUK1o00Wbjb-wFch`$zi%)C0B; z96UJCu>XW1a6OFyf(b?93IV3kU%l!d84N`t(NJ73AcgV6ub%F{Az- zQNX^3n>0|=|A!Y)!YJH|0Q#FS?nHs2(0?>8C9KbAx^kp4Kz0D~Het01s7z;Smvpe-P9_h?{~_-E+; z-Ua|59CzRP4f|&cc6Bzfd2HcKDkjDzq;BH{{8Ed6k_xFgI=TYyfcyZSXCS2LVCe|_ 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Analogamente, se $L < 0$, $(x_n)$ è negativa definitivamente. \end{theorem} \begin{proof} - Senza perdita di generalità si pone $L > 0$. Allora esiste sicuramente un intorno $I$ di $L$ tale che ogni suo elemento è positivo (e.g.~$I = [\frac{L}{2}, \frac{3L}{2}]$, se $L \in \RR$, altrimenti $[a, \infty]$ con $a > 0$ se $L = +\infty$). Dal momento che $x_n \tendston L$, $\exists n_k \mid n \geq n_k \implies x_n \in I$, - ossia, in particolare, $n \geq n_k \implies x_n > 0$, da cui la tesi. + Senza perdita di generalità si pone $L > 0$. Allora esiste sicuramente un intorno $I$ di $L$ tale che ogni suo elemento è positivo (e.g.~$I = \left[L - \frac{L}{2}, L + \frac{L}{2}\right]$, se $L \in \RR$, altrimenti $[a, \infty]$ con $a > 0$ se $L = +\infty$). Dal momento che $x_n \tendston L$, $\exists n_k \mid n \geq n_k \implies x_n \in I$, + ossia, in particolare, $n \geq n_k \implies x_n > 0$, da cui la tesi. Analogamente per il caso di $L < 0$. \end{proof} - \begin{proposition} + \begin{theorem} (della permanenza del segno, per le funzioni) Sia $f : X \to \RRbar$ e sia $\xbar$ un punto di accumulazione di $X$. Se $\lim_{x \to \xbar} f(x) = L > 0$, allora $\exists J$ intorno non vuoto di $\xbar$ tale che $f(x) > 0$ $\forall x \in J \cap X \setminus \{\xbar\}$. - \end{proposition} + Analogamente se $L < 0$, $\exists J$ intorno non vuoto di $\xbar$ tale che $f(x) < 0$ $\forall x \in J \cap X \setminus \{\xbar\}$. + \end{theorem} \begin{proof} - Analogamente a come visto per il teorema del segno, si pone $L > 0$. Allora esiste sicuramente un intorno $I$ di $L$ tale che ogni suo elemento è positivo. Poiché $\lim_{x \to \xbar} f(x) = L > 0$, deve esistere un intorno $J$ di - $\xbar$ tale che $f(J \cap X \setminus \{\xbar\}) \subseteq I$. In particolare, $J \cap X \setminus \{\xbar\}$ non - è mai vuoto, dal momento che $\xbar$ è un punto di accumulazione di $X$, e vale che $f(x) > 0$ $\forall x \in J \cap X \setminus \{\xbar\}$ (dal momento che $f(x) \in I$, che ha tutti elementi positivi), da cui la tesi. + Senza perdita di generalità si pone $L > 0$. + Similmente a come visto per l'analogo teorema per le successioni, deve esiste sicuramente un intorno $I$ di $L$ tale che ogni suo elemento è positivo. Poiché $\lim_{x \to \xbar} f(x) = L > 0$, deve in particolare esistere un intorno $J$ di + $\xbar$ tale che $f(J \cap X \setminus \{\xbar\}) \subseteq I$. Inoltre, $J \cap X \setminus \{\xbar\}$ non può + essere mai vuoto, dal momento che $\xbar$ è un punto di accumulazione di $X$. Infine vale anche che $f(x) > 0$ $\forall x \in J \cap X \setminus \{\xbar\}$ (dal momento che $f(x) \in I$, che ha tutti elementi positivi). + Si conclude allora che $J$ è l'intorno desiderato, da cui la tesi. Analogamente per il caso di $L < 0$. \end{proof} - \begin{theorem} (degli zeri) Dati $I = [a, b]$ e - $f : I \to \RRbar$ continua tale che $f(a) f(b) < 0$ (i.e.~sono discordi), allora $\exists c \in (a, b) \mid f(c) = 0$. + \begin{theorem} (degli zeri) Sia $I = [a, b]$ e sia + $f : I \to \RRbar$ continua tale che $f(a) f(b) < 0$ (i.e.~$f(a)$ e $f(b)$ sono discordi). Allora $\exists c \in (a, b) \mid f(c) = 0$. \end{theorem} \begin{proof} - Senza alcuna perdita di generalità si pone $f(a) < 0 < f(b)$ (il caso $f(a) > 0 > f(b)$ è - infine dimostrato considerando $g(x) = -f(x)$). Si definisce allora l'insieme $E$ in modo tale che: + Senza alcuna perdita di generalità si pone $f(a) < 0 < f(b)$ (il caso $f(a) > 0 > f(b)$ si dimostra + considerando $g = -f$). Si definisce allora l'insieme $E$ in modo tale che: - \[ E = \{ a \in I \mid f(a) < 0 \}. \] + \[ E = \{ x \in I \mid f(x) < 0 \}. \] \vskip 0.05in - Si osserva che $E \neq \emptyset$, dacché $a \in E$. Per la completezza dei numeri reali, - $E$ ammette un estremo superiore $\xbar := \sup E$. Sia $(x_n) \subseteq E$ una successione + Si osserva che $E \neq \emptyset$, dacché $a \in E$. Allora, per la completezza dei numeri reali, + $E$ ammette un estremo superiore $\xbar := \sup E$. Sia dunque $(x_n) \subseteq E \setminus \{\xbar\}$ una successione tale che $x_n \tendston \xbar$: poiché $f$ è continua in $\xbar$, $\lim_{x \to \xbar} f(x) = f(\xbar) \implies - f(x_n) \tendston f(\xbar)$. Allora, poiché $f(x_n) < 0$ $\forall n \in \NN$, $f(\xbar) \leq 0$ (se così non fosse + f(x_n) \tendston f(\xbar)$. Dal momento che $f(x_n) < 0$ $\forall n \in \NN$, $f(\xbar) \leq 0$ (se così non fosse $f(x_n)$ dovrebbe essere definitivamente positiva per il teorema della permanenza del segno, ma questo è assurdo dacché $x_n \in E$ $\forall n \in \NN$, \Lightning). \\ Sia ora $(y_n) \in I$ una successione tale che $y_n \tendston \xbar$ e che $y_n > \xbar$ $\forall n \in \NN$ (questo è sempre possibile dal momento che $\xbar \neq b \impliedby f(\xbar) \leq 0$). Allora, poiché $y_n > \xbar = \sup E$, $y_n$ non appartiene ad $E$, e quindi deve valere che $f(y_n) > 0$. Si conclude - allora, per il teorema della permanenza del segno, che $f(\xbar) \geq 0$, e quindi che $f(\xbar) = 0$, da cui - la tesi. + allora, per il teorema della permanenza del segno, che $f(\xbar) \geq 0$. Pertanto $f(\xbar) \leq 0$ e $f(\xbar) \geq 0$ $\implies f(\xbar) = 0$, da cui la tesi. \end{proof} \begin{proof}[Dimostrazione alternativa] (metodo di bisezione per la ricerca degli zeri) - Come prima, senza alcuna perdita di generalità, si pone $f(a) < 0 < f(b)$. Si ponga $x_0 = \frac{a+b}{2}$, + Come prima, senza alcuna perdita di generalità, si pone $f(a) < 0 < f(b)$. Si ponga $x_0 = \frac{a+b}{2} \in I$ e $I_0 = (a, b)$. Se $f(x_0) = 0$, allora il teorema è dimostrato. Altrimenti, $f(x_0) > 0$ o $f(x_0) < 0$. Nel primo caso, - si consideri $I_1 = (a, x_0)$, altrimenti si ponga $I_1 = (x_0, b)$. Si riapplichi allora l'algoritmo con $a := \inf I_1$ e $b := \sup I_1$, definendo la successione $(x_n)$ e gli intervalli $I_n$ per ogni passo $n$ dell'algoritmo. \\ + si consideri $I_1 = (a, x_0)$, altrimenti si ponga $I_1 = (x_0, b)$. Si riapplichi allora l'algoritmo con $a := \inf I_1$ e $b := \sup I_1$, definendo le due successioni $(x_n)$ e $I_n$ per ogni passo $n$ dell'algoritmo. \\ Se la successione $(x_n)$ è finita, allora $\exists n \mid f(x_n) = 0$, e quindi il teorema è dimostrato. Altrimenti, si osservi che la successione degli intervalli è decrescente, e che $\abs{I_n} = \frac{b-a}{2^n} \tendston 0$: allora, poiché $x_n \in I_n$ $\forall n \in \NN$, $(x_n)$ ammette limite. In particolare, $I_n \tendston \{c\}$, - e quindi $x_n \tendston c \in I_0$. Siano $a_n$, $b_n$ le successioni tali che $I_n = (a_n, b_n)$ $\forall n \in \NN$. - Vale in particolare che $a_n, b_n \tendston c$. Allora, per la continuità di $f$ su $(a, b)$, vale che + e quindi $x_n \tendston c \in I_0$. Siano $a_n$, $b_n$ le successioni di numeri reali tali che $I_n = (a_n, b_n)$ $\forall n \in \NN$. + Vale in particolare che $a_n$, $b_n \tendston c$. Allora, per la continuità di $f$ su $(a, b)$, vale che $\lim_{n \to \infty} f(a_n) = f(c)$ e che $\lim_{n \to \infty} f(b_n) = f(c)$: poiché ogni elemento di $(a_n)$ è per costruzione tale che $f(a_n) < 0$, deve valere che $f(x) \leq 0$ per il teorema della permanenza del segno; analogamente deve valere per costruzione di $(b_n)$ che $f(c) \geq 0$. Si conclude allora che $f(c) = 0$, da cui la tesi. \end{proof} - \begin{corollary} (dei valori intermedi) Dati $I = (a, b)$ e + \begin{theorem} (dei valori intermedi) Dati $I = (a, b)$ e $f : I \to \RRbar$ continua, allora $y_1$, $y_2 \in f(I) \implies [y_1, y_2] \subseteq f(I)$ (ossia $f$ assume tutti i valori compresi tra $y_1$ e $y_2$; e quindi $f(I)$ è un insieme convesso di $\RR$). - \end{corollary} + \end{theorem} \begin{proof} Supponiamo $y_1 < y_2$: poiché $y_1$, $y_2$ appartengono già a $f(I)$, è sufficiente mostrare che anche ogni $y \in (y_1, y_2)$ appartiene a $f(I)$. Dal momento che $y_1$, $y_2 \in f(I)$, $\exists x_1$, $x_2 \in I \mid f(x_1) = y_1$ e $f(x_2) = y_2$. Si consideri allora $g : I \to \RRbar$ tale che @@ -507,7 +507,7 @@ \end{proposition} \begin{proof} - La dimostrazione del fatto che gli intervalli siano convessi è banale. Si dimostra piuttosto + La dimostrazione del fatto che gli intervalli siano insiemi convessi è banale. Si dimostra piuttosto che ogni insieme convesso di $\RR$ è un intervallo. Sia $A$ dunque un insieme convesso di $\RR$, e si considerino $a := \inf A$ e $b := \sup A$. Sia $x \in (a, b)$. Se non esistesse un punto $c \in A$ tale che $a < c < x$, $x$ sarebbe un estremo inferiore di $A$, \Lightning. Pertanto tale punto $c$ esiste. @@ -515,7 +515,7 @@ $[c, d] \subseteq A$, e in particolare $x \in A$. Pertanto vale che $(a, b) \subseteq A$. Poiché $a$ e $b$ sono, rispettivamente, estremo inferiore e superiore di $A$, non possono esistere altri punti non appartenenti a $[a, b]$, ma appartenenti ad $A$. Quindi $A$ può variare a seconda dell'appartenenza - o meno di questi estremi tra questi insiemi: + o meno di questi estremi nei seguenti modi: \begin{enumerate}[(i)] \item $A = (a, b)$, se $a$, $b \notin A$, @@ -540,8 +540,8 @@ \begin{proof} Ci si limita a dimostrare l'esistenza del minimo, dacché l'esistenza - del massimo segue dal considerare $g = -f$. Sia $m := \inf f(I)$. - Esiste allora una successione $(y_n) \subseteq f(I)$ tale che + del massimo segue dal considerare $g = -f$. Sia $m := \inf f(I)$. Esiste + allora una successione $(y_n) \subseteq f(I)$ tale che $y_n \tendston m$. Poiché $y_n \in f(I)$, $\exists x_n \in I \mid y_n = f(x_n)$. Per il teorema di Bolzano-Weierstrass, $\exists \, (x_{n_k}) \subseteq I$ sottosuccessione convergente, ossia tale che @@ -561,7 +561,7 @@ \end{remark} \begin{remark} (algoritmo di ricerca dei massimi e dei minimi) Sia $f : I \to \RRbar$ la funzione continua di cui si - ricerca i massimi e i minimi. Si ipotizzi\footnote{Non è infatti sempre possibile considerarne un'estensione continua (e.g.~$\sin(\frac1{x})$, il seno del topologo); ciò accade qualora non esista almeno uno dei limiti negli estremi dell'intervallo di $I$.} di poter considerare $\tilde{f} : \overline{I} \to \RRbar$, ossia l'estensione + ricerca i massimi e i minimi. Si ipotizzi\footnote{Non è infatti sempre possibile considerarne un'estensione continua (e.g.~$\sin\left(\frac1{x}\right)$, il seno del topologo); ciò accade qualora non esista almeno uno dei limiti negli estremi dell'intervallo di $I$.} di poter considerare $\tilde{f} : \overline{I} \to \RRbar$, ossia l'estensione continua di $f$. Allora, poiché $\tilde{f}$ è continua ed è definita su un intervallo chiuso, per Weierstrass ammette un massimo e un minimo. Preso per esempio il minimo, esso potrebbe essere un