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\begin{proposition} Dati $f : X \to \RRbar$, $\xbar$ punto di accumulazione di $X$ - tale che $\forall \, (x_n) \subseteq X \setminus \{\xbar\}$ vale che - $f(x_n)$ converge. Allora il limite di $f(x_n)$ è sempre lo stesso. - \end{exercise} + tale che $\forall \, (x_n) \subseteq X \setminus \{\xbar\} \mid x_n \tendston \xbar$ vale che + $f(x_n)$ converge. Allora il limite di $f(x_n)$ è sempre lo stesso, indipendentemente + dalla scelta di $(x_n)$. + \end{proposition} - \begin{exercise} + \begin{proof} + Siano per assurdo $(x_n), (y_n) \subseteq X \setminus \{\xbar\}$ due successioni tali che + $x_n, y_n \tendston \xbar$ e che $f(x_n) \tendston L$ e $f(y_n) \tendston G$ con $L \neq G$. Si + costruisce allora la successione $(z_n) \subseteq X \setminus \{\xbar\}$ nel seguente modo: + + \[ z_n = \system{x_{\frac{n}{2}} & \text{se } n \text{ è pari}, \\ y_{\frac{n-1}{2}} & \text{altrimenti},} \] + + \vskip 0.05in + + ossia unendo le due successioni $(x_n)$ e $(y_n)$ in modo tale che agli indici pari corrispondano gli + elementi di $x_n$ e a quelli dispari quelli di $y_n$. \\ + + Si mostra che $z_n \tendston \xbar$. Sia $I$ un intorno di $\xbar$. Allora, dal momento che + $(x_n), (y_n) \tendston \xbar$, esistono sicuramente due + $n_x, n_y \in \NN$ tali che $n \geq n_x \implies x_n \in I$ e $n \geq n_y \implies y_n \in I$. Pertanto, + detto $n_k = \max\{n_x, n_y\}$, $n \geq n_k \implies x_n, y_n \in I$, ossia che per $n \geq 2 n_k$, + $z_n \in I$. Si conclude allora che $(z_n) \tendston \xbar$. \\ + + Tuttavia $f(z_n)$ non può convergere a nessun limite, dal momento che le due sottosuccessioni + $f(x_n)$ e $f(y_n)$ convergono a valori distinti ed il limite deve essere unico. L'esistenza di + tale successione contraddice allora l'ipotesi, \Lightning. + \end{proof} + + \begin{proposition} Data $(x_n) \subseteq \RR$, definisco $f : \NN \to \RRbar$ tale che $f(n) := x_n$, $\forall n \in \NN$. Allora $f(n) \tendston L \iff x_n \tendston L$. - \end{exercise} + \end{proposition} + + \begin{proof} Si dimostrano le due implicazioni separatamente. \\ + + \rightproof Sia $I$ un intorno di $L$. Allora, poiché $f(n) \tendston L$, + esiste un intorno $J = [a, \infty]$ tale che $f(J \cap \NN \setminus \{\infty\}) \subseteq I$. + Poiché $\infty$ è un punto di accumulazione di $\NN$, $A = J \cap \NN \setminus \{\infty\}$ non è mai + vuoto. Inoltre, poiché $A \subseteq \NN$, $A$ ammette un minimo\footnote{Non è in realtà necessario che + si consideri il minimo di tale insieme, occorre semplicemente che $A$ sia non vuoto.}, detto $m$. + Vale in particolare che + $f(n) \in I$, $\forall n \geq m$, e quindi che $x_n \in I$, $\forall n \geq m$, ossia che $x_n \tendston L$. \\ + + \leftproof Sia $I$ un intorno di $L$. Dal momento che $x_n \tendston L$, $\exists n_k \in \NN \mid n \geq n_k \implies + x_n \in I$. Allora, detto $J = [n_k, \infty]$, vale che $f(J \cap \NN \setminus \{\infty\}) \subseteq I$, ossia + che $f(n) \tendston L$. + \end{proof} \begin{proposition} Siano $f : X \to \RRbar$, $\xbar \in X$ punto di accumulazione di $X$. Allora sono fatti equivalenti i seguenti: \begin{enumerate}[(i)] - \item $f(x) \tendsto{\xbar} L$, + \item $f(x) \tendsto{\xbar} f(\xbar)$, \item $f$ è continua in $\xbar$. \end{enumerate} \end{proposition} \begin{proof} - Si dimostrano le due implicazioni separatamente. + Sia $I$ un intorno di $f(\xbar)$. Dal momento che $\xbar$ è un punto di accumulazione, si ricava allora da + entrambe le ipotesi che esiste un intorno $J$ di $f(\xbar)$ tale che + $f(J \cap X \setminus \{\xbar\}) \subseteq I$, e quindi, per definizione, la tesi. \end{proof} \begin{remark} @@ -54,7 +95,7 @@ continua). \end{remark} - \begin{exercise} + \begin{proposition} Siano $f : X \to \RR$ e $\xbar$ punto di accumulazione di $X$. Siano $L \in \RRbar$ e $\tilde{f} : X \cup \{\xbar\} \to \RRbar$ tale che: @@ -67,38 +108,88 @@ \vskip 0.05in Allora $f(x) \tendsto{\xbar} L \iff \tilde{f}$ è continua in $\xbar$. - \end{exercise} + \end{proposition} + + \begin{proof} + Si dimostrano le due implicazioni separatamente. \\ + + \rightproof Sia $I$ un intorno di $L$. Si ricava allora dalle ipotesi che esiste sempre un intorno + $J$ di $\xbar$ tale che $f(\underbrace{J \cap X \setminus \{\xbar\}}_{A}) \subseteq I$. Dal momento che $\xbar + \notin A$, si deduce che $f(J \cap X \setminus \{\xbar\}) = \tilde{f}(J \cap X \setminus \{\xbar\}) \subseteq I$, + ossia che $\tilde{f}$ è continua in $\xbar$. \\ + + \leftproof Sia $I$ un intorno di $L$. Poiché $\tilde{f}$ è continua in $\xbar$, esiste un intorno $J$ di $\xbar$ + tale che $\tilde{f}(\underbrace{J \cap (X \cup \{\xbar\}) \setminus \{\xbar\}}_{A}) \subseteq I$. Poiché $\xbar \notin A$ e $\xbar$ è punto di accumulazione, si deduce che $I \supseteq \tilde{f}(J \cap (X \cup \{\xbar\}) \setminus \{\xbar\}) + = f(J \cap (X \cup \{\xbar\}) \setminus \{\xbar\}) \supseteq f(J \cap X \setminus \{\xbar\})$, e quindi che + $f(x) \tendsto{\xbar} L$. + \end{proof} \begin{remark} - Tutte le funzioni elementari (e.g.~$\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, $\abs{x}$, polinomi) sono funzioni continue nel loro insieme + Tutte le funzioni elementari (e.g.~$\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, $\abs{x}$, $x^a$) sono funzioni continue nel loro insieme di definizione. \end{remark} \begin{proposition} - Date $f : X \to Y \subseteq \RRbar$ e $g : Y \to \RRbar$. Sia + Siano $f : X \to Y \subseteq \RRbar$ e $g : Y \to \RRbar$ e sia $\xbar \in X$. Sia $f$ continua in $\xbar$ e sia $g$ continua in $f(\xbar)$. Allora $g \circ f$ è continua in $\xbar$. \end{proposition} \begin{proof} Sia $I$ un intorno di $z = g(f(\xbar))$. Allora, poiché $g$ è continua - in $f(\xbar)$, $\exists J$ intorno di $f(\xbar)$ $\mid g(J) \subseteq + in $f(\xbar)$, $\exists J$ intorno di $f(\xbar)$ $\mid g(J \cap Y \setminus \{f(\xbar)\}) \subseteq I$. Tuttavia, poiché $f$ è continua in $\xbar$, $\exists K$ intorno - di $\xbar$ $\mid f(K) \subseteq J$, da cui si conclude che - $g(f(K)) \subseteq g(J) \subseteq I$. + di $\xbar$ $\mid f(K \cap X \setminus \{\xbar\}) \subseteq J$, da cui si conclude che + $g(f(K \cap X \setminus \{\xbar\})) \subseteq I$, dacché $\forall x \in K \cap X \setminus \{\xbar\}$, + o $f(x) = f(\xbar)$, e quindi $g(f(x)) = z$ chiaramente appartiene a $I$, o altrimenti + $f(x) \in J \cap Y \setminus \{f(\xbar)\} \implies g(f(x)) \in g(J \cap Y \setminus \{f(\xbar)\}) \subseteq I$. \end{proof} - \begin{proposition} + \begin{theorem} Sia $f : X \to Y \subseteq \RRbar$, sia $\xbar$ punto di accumulazione di $X$ tale che $f(x) \tendsto{\xbar} \ybar$. - Se $\ybar$ è un punto di accumulazione di $Y$, $\ybar \in Y \implies - g$ continua in $\ybar$ e $g : Y \to \RRbar$ - è tale che $g(y) \tendstoy{\ybar} \zbar$, allora + Se $\ybar$ è un punto di accumulazione di $Y$ e $g : Y \to \RRbar$ + è tale che $\ybar \in Y \implies + g$ continua in $\ybar$ e $g(y) \tendstoy{\ybar} \zbar$, allora $g(f(x)) \tendsto{\xbar} \zbar$. - \end{proposition} + \end{theorem} \begin{proof} + Siano $\tilde{f} : X \cup \{\xbar\}$, $\tilde{g} : Y \cup \{\ybar\}$ due funzioni costruite nel seguente + modo: + + \[ \tilde{f}(x) = \begin{cases} + \ybar & \text{se } x = \xbar, \\ + f(x) & \text{altrimenti}, + \end{cases} \qquad + \tilde{g}(y) = \begin{cases} + \zbar & \text{se } y = \ybar, \\ + g(y) & \text{altrimenti}. + \end{cases} \] + + Poiché $f(x) \tendsto{\xbar} \ybar$ e $\xbar$ è un punto di accumulazione di $X$, per una proposizione precedente, $\tilde{f}$ è continua in $\xbar$. Analogamente $\tilde{g}$ è continua in $\ybar$. Dal momento che + vale che $\tilde{f}(\xbar) = \ybar$, per la proposizione precedente $\tilde{g} \circ \tilde{f}$ è continua in + $\xbar$, e dunque $\lim_{x \to \xbar} \tilde{g}(\tilde{f}(x)) = \tilde{g}(\tilde{f}(\xbar)) = \zbar$. \\ + + Si consideri adesso la funzione $\widetilde{g \circ f} : X \to \RRbar$ definita nel seguente modo: + + \[ \widetilde{g \circ f}(x) = \begin{cases} + \zbar & \text{se } x = \xbar, \\ + g(f(x)) & \text{altrimenti}. + \end{cases} \] + + Si mostra che $\widetilde{g \circ f} = \tilde{g} \circ \tilde{f}$. Se $x = \xbar$, chiaramente + $\widetilde{g \circ f}(x) = \zbar = \tilde{g}(\tilde{f}(\xbar))$. Se $x \neq \xbar$, si + considera il caso in cui $\tilde{f}(x) = f(x)$ è uguale a $\ybar$ ed il caso in cui non vi è + uguale. \\ + + Se $\tilde{f}(x) \neq \ybar$, $\tilde{g}(\tilde{f}(x)) = \tilde{g}(f(x)) \overbrace{=}^{f(x) \neq \ybar} g(f(x)) = \widetilde{g \circ f}(x)$. Se invece + $\tilde{f}(x) = \ybar$, $\ybar \in Y$, e quindi $g$ è continua in $\ybar$, da cui necessariamente + deriva che $g(\ybar) = \zbar$. Allora $\widetilde{g \circ f}(x) = g(f(x)) = g(\ybar) = \zbar = \tilde{g}(\tilde{f}(\xbar))$. \\ + Si conclude allora che $\widetilde{g \circ f} = \tilde{g} \circ \tilde{f}$, e + quindi che $\widetilde{g \circ f}$ è continua in $\xbar$. Pertanto, + dalla proposizione precedente, $g(f(x)) \tendsto{\xbar} \zbar$. \end{proof} \begin{exercise} diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index f6feaea..ba56557 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -10,6 +10,9 @@ \usepackage{marvosym} \usepackage{float} \usepackage{enumerate} +\usepackage{scalerel} +\usepackage{stackengine} +\usepackage{wasysym} \hfuzz=\maxdimen \tolerance=10000 @@ -102,6 +105,9 @@ \newcommand{\zeroset}{\{0\}} \newcommand{\setminuszero}{\setminus \{0\}} +\newenvironment{solution} +{\textit{Soluzione.}\,} + \theoremstyle{definition} \let\abstract\undefined