From fc3fa6bd60ab34ab2dd0d895e1771f1d46285725 Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 24 Mar 2023 23:55:51 +0100 Subject: [PATCH] feat(analisi): aggiorna gli appunti del 24/03/2023 --- .../main.tex | 2 +- .../main.pdf | Bin 216113 -> 223842 bytes .../main.tex | 74 +++++++++++++++--- 3 files changed, 65 insertions(+), 11 deletions(-) diff --git a/Analisi I/Parte teorica/2023-03-22, Limiti di funzioni e funzioni continue/main.tex b/Analisi I/Parte teorica/2023-03-22, Limiti di funzioni e funzioni continue/main.tex index e896c66..6f49bc4 100644 --- a/Analisi I/Parte teorica/2023-03-22, Limiti di funzioni e funzioni continue/main.tex +++ b/Analisi I/Parte teorica/2023-03-22, Limiti di funzioni e funzioni continue/main.tex @@ -115,7 +115,7 @@ tale che $n \geq n_k \implies f(x_n) \in I$, ossia $f(x_n) \tendston L$. \\ \leftproof Si ponga per assurdo che $\lim_{x \to \xbar} f(x) \neq L$. Allora esiste almeno - un intorno $I$ di $L$ tale per cui non esista alcun intorno $J$ di $\xbar \mid f(J \cap X \setminus \{\xbar\}) \subseteq I$. Si consideri adesso il caso $\xbar \in \RR$ ed il suo intorno $J_n = [\xbar - \frac{1}{n}, \xbar + \frac{1}{n}]$: per ogni $n$ si può estrarre $J_n$ un $k \in X \setminus \{\xbar\}$ + un intorno $I$ di $L$ tale per cui non esista alcun intorno $J$ di $\xbar \mid f(J \cap X \setminus \{\xbar\}) \subseteq I$. Si consideri adesso il caso $\xbar \in \RR$ ed il suo intorno $J_n = [\xbar - \frac{1}{n}, \xbar + \frac{1}{n}]$: da ogni $J_n$ si può estrarre un $k \in X \setminus \{\xbar\}$ (infatti $\xbar$ è un punto di accumulazione), tale che $f(k) \notin I$. Si ponga allora $x_n = k$. Dal momento che $J_n \tendston \{\xbar\}$, $x_n \tendston \xbar$. Allo stesso tempo, per $n \to \infty$, $f(x_n)$ non può tendere a $L$, dacché per costruzione $f(x_n)$ non appartiene all'intorno $I$. Tuttavia ciò contraddice l'ipotesi, e quindi $\lim_{x \to \xbar} f(x) = L$. \\ diff --git a/Analisi I/Parte teorica/2023-03-23, 24, Proprietà principali della continuità e dei limiti di funzione/main.pdf b/Analisi I/Parte teorica/2023-03-23, 24, Proprietà principali della continuità e dei limiti di funzione/main.pdf index 16d3db05ba82e8f8ad916a0f417a4cf96cddc547..89976d634ed5013e1fb1eba0d871888a9c4f7b9d 100644 GIT binary patch delta 54794 zcmY(pbC4!J^zPZVZQHgvZA{zNwC%TTzim(3J#E{zZQI`O-MzoNyMH8Ad2({!t z^_61uc17xVt!X>k@rRp|GKD#FHMSV=moZY=;`e$zF2ByRS~Lv-wUWa$PK$MySK%?V zQpxif!zuJxU0qY#u@(o0+uyfOFa2g=JmTWwl4VY7A*it=L;89NPo>Stb!e zlL_{d=7Q-HZ6uiwqUiwLx$}NKFQMal#YsUK#R^56{@XsQ#>e&KmQN2cmlP_rS{vGk zfsj7EpHK;aF-=grt>A=E@}=!#e>p{e#{JdoM2`@q*(k-{<=mE)V!Ex{LmrQv%%9E- z8W{%lwL!ECGhjHPBE2*M5H~nky9UxsU`{IbyOBVNqOkDe>>Xh2-=HJE((p%M&)yXo z@n7K@pMOvK@WqvH!&hsLzusi&gwk6>IH!=jZ;t^|l#vKQa5CTXBWo$%5Y@#2n!iZN zY!RR+SptPlOKv4oq9BjKTAm_h)`;mVM;1L)8-_-X8gV&?L(9V%xgB`31ib~GEN0Z5 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definitivamente. Analogamente, se $L < 0$, + $(x_n)$ è strettamente positiva definitivamente. Analogamente, se $L < 0$, $(x_n)$ è negativa definitivamente. \end{theorem} \begin{proof} + Se $L > 0$, allora esiste sicuramente un intorno $I$ di $L$ tale che ogni suo elemento è positivo (e.g.~$I = [\frac{L}{2}, \frac{3L}{2}]$). 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. \end{proof} \begin{theorem} (degli zeri) Dati $I = [a, b]$ e @@ -300,17 +343,28 @@ \end{theorem} \begin{proof} - Prendo $E$ insieme dei punti in cui la funzione è negativa e - dimostro che il $\sup$ di questo insieme è proprio $0$ (esiste - per la completezza di $\RR$ e mostro che $f(\xbar) = 0$). Mostro - quest'ultimo concetto facendo vedere che è sia positivo che - negativo dalle ipotesi (si può sempre approssimare l'estremo - superiore di $E$ con i suoi elementi). $\xbar$ si può approssimare - anche da destra (essendo il $\sup$ di $E$, deve valere che le - successioni da destra sono positive) + 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: + + \[ E = \{ a \in I \mid f(a) < 0 \}. \] + + \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 + 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 + $f(x_n)$ 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. \end{proof} - \begin{corollary} (dei valori intermedi) Dati $I = (a, b)]$ e + \begin{corollary} (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 $y_2$).