From bf3a13b4e31a2440c8829d9dd55a3e71708ac5fe Mon Sep 17 00:00:00 2001 From: Gabriel Videtta Date: Fri, 24 Mar 2023 19:05:20 +0100 Subject: [PATCH] feat: aggiunge le basi degli appunti di 24/03/2023 --- .../main.pdf | Bin 0 -> 216113 bytes .../main.tex | 93 +++++++++++++++++- .../main.pdf | Bin 206349 -> 0 bytes .../main.pdf | Bin 199907 -> 199907 bytes .../2023-03-24/main.pdf | Bin 0 -> 71990 bytes .../2023-03-24/main.tex | 26 +++++ tex/latex/style/personal_commands.sty | 2 +- 7 files changed, 117 insertions(+), 4 deletions(-) create mode 100644 Analisi I/Parte teorica/2023-03-23, 24, Proprietà principali della continuità e dei limiti di funzione/main.pdf rename Analisi I/Parte teorica/{2023-03-23, Proprietà principali della continuità e dei limiti di funzione => 2023-03-23, 24, Proprietà principali della continuità e dei limiti di funzione}/main.tex (71%) delete mode 100644 Analisi I/Parte teorica/2023-03-23, Proprietà principali della continuità e dei limiti di funzione/main.pdf 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Proprietà principali della continuità e dei limiti di funzione/main.tex +++ b/Analisi I/Parte teorica/2023-03-23, 24, Proprietà principali della continuità e dei limiti di funzione/main.tex @@ -4,7 +4,7 @@ \title{\textbf{Note del corso di Analisi Matematica 1}} \author{Gabriel Antonio Videtta} -\date{21 marzo 2023} +\date{23 e 24 marzo 2023} \begin{document} @@ -206,10 +206,10 @@ \end{proposition} \begin{proof} - Sia $f := f_1 + f_2$. + Si dimostrano i due punti separatamente. \begin{enumerate}[(i)] - \item Poiché $f_1, f_2$ sono continue in $\xbar$, + \item Sia $f := f_1 + f_2$. Poiché $f_1, f_2$ sono continue in $\xbar$, $\forall \eps > 0$, $\exists \delta > 0 \mid \abs{x - \xbar} < \delta \implies \abs{f_1(x) - f_1(\xbar)}, \abs{f_2(x) - f_2(\xbar)} \leq \eps$ (per ogni $\eps > 0$, si prende $\delta = \min\{\delta_1, \delta_2\}$, ossia il minimo delle semilunghezze degli intorni di $\xbar$). Allora $\abs{f(x) - f(\xbar)} \leq @@ -217,6 +217,18 @@ Si conclude dunque che $\forall \eps > 0$, $\exists \delta > 0 \mid \abs{f(x) - f(\xbar)} \leq 2\eps$, e quindi, poiché $2\eps \tends{\eps \to 0} 0$, che $f$ è continua in $\xbar$. + + \item Dal momento che $f_1, f_2$ sono continue in $\xbar$, + $\forall \eps > 0$, $\exists \delta > 0$ tale che $\abs{x - \xbar} < \delta \implies \abs{f_1(x) - f_1(\xbar)} < \eps, \abs{f_2(x) + - f_2(\xbar)} < \eps$ (vale lo stesso ragionamento del punto + (i)). Allora $f_1(x) = f_1(\xbar) + e_1$ e + $f_2(x) = f_2(\xbar) + e_2$, con $\abs{e_1}, \abs{e_2} < \eps$. + Dunque $f_1(x)f_2(x) = f_1(\xbar)f_2(\xbar) + \underbrace{e_1 f_2(\xbar) + + e_2 f_1(\xbar) + e_1 e_2}_e$. In particolare, per la + disuguaglianza triangolare, $\abs{e} \leq \abs{e_1 f_2(\xbar)} + + \abs{e_2 f_1(\xbar)} + \abs{e_1 e_2} \leq \underbrace{\eps \abs{f_2(\xbar)} + + \eps \abs{f_1(\xbar)} + \eps^2}_{\eps'}$. Poiché $\eps' \tends{\eps \to 0^+} 0$, si conclude che $\abs{f_1(x) f_2(x) - f_1(\xbar) f_2(\xbar)} = \abs{e} \leq \eps' \implies f_1(x)f_2(x)$ continua + in $\xbar$. \end{enumerate} \end{proof} @@ -232,4 +244,79 @@ \end{enumerate} \end{proposition} + \begin{definition} + (intorno destro e sinistro) Se $\xbar \in \RR$, si dicono + \textbf{intorni destri} gli intervalli della forma $[\xbar, \xbar + \eps]$ con + $\eps > 0$. Analogamente, gli \textbf{intorni sinistri} sono gli + intervalli della forma $[\xbar - \eps, \xbar]$. + \end{definition} + + \begin{definition} + (punto di accumulazione destro e sinistro) Sia $\xbar \in X$. + Si dice che $\xbar$ è un \textbf{punto di accumulazione destro} + di $X$ se $\forall I$ intorno destro di $\xbar$, $I \cap X \setminus \{\xbar\} \neq \emptyset$. Analogamente si dice \textbf{punto di + accumulazione sinistro} di $X$ se è tale per gli intorni sinistri. + \end{definition} + + \begin{definition} + (limite destro e sinistro) Sia $\xbar$ un punto di accumulazione + destro di $X$. Allora $\lim_{x \to \xbar^+} f(x) = L \defiff \forall I$ + intorno di $L$, $\exists J$ intorno destro di $\xbar$ tale che + $f(J \cap X \setminus \{\xbar\}) \subseteq I$. Analogamente si definisce + il limite sinistro. + \end{definition} + + \begin{definition} + (continuità destra e sinistra) Sia $\xbar \in X$. Allora $f$ è continua + a destra in $\xbar$ se e solo se $\forall I$ intorno di $f(\xbar)$, + $\exists J$ intorno destro di $\xbar$ tale che $f(J \cap X \setminus \{\xbar\}) \subseteq I$. Analogamente si definisce la continuità + a sinistra di $f$. + \end{definition} + + %TODO: aggiungere osservazioni sulla continuità destra e sinistra. + + \begin{proposition} + Sia $f : X \to \RRbar$ monotona e sia $\xbar$ un punto di + accumulazione destro di $X$. Allora esiste $\lim_{x \to \xbar^+} f(x)$. + Analogamente esiste da sinistra se $\xbar$ è un punto di + accumulazione sinistro di $X$. + \end{proposition} + + %TODO: aggiungere funzione discontinua in ogni punto di R. + + %TODO: l'insieme dei punti di discontinuità per una funzione monotona è al più numerabile (hint: punto medio). + + \begin{theorem} (della permanenza del segno) + Data $(x_n) \subseteq \RR$ tale che $x_n \tendston L > 0$, allora + $(x_n)$ è positiva definitivamente. Analogamente, se $L < 0$, + $(x_n)$ è negativa definitivamente. + \end{theorem} + + \begin{proof} + \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 $\exists c \in (a, b) \mid f(c) = 0$. + \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) + \end{proof} + + \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 $y_2$). + \end{corollary} + + \begin{proof} + + \end{proof} \end{document} diff --git a/Analisi I/Parte teorica/2023-03-23, Proprietà principali della continuità e dei limiti di funzione/main.pdf b/Analisi I/Parte teorica/2023-03-23, Proprietà principali della continuità e dei limiti di funzione/main.pdf deleted file mode 100644 index de0ac5df0170d3d4d6170f8ee3a7fba71912becf..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 206349 zcmb@u1z6NUw>V5UNQZ!Q=dx^ogoH?UcOxa;p&%^^ij;tagp{I4D4j|pA&qnjh)Re5 z-|pfo@4fH6_r2f!e4d@1J#*&F%sFRHj5Ba($}0$f1R(?*gCiZ|1Yl-iW;bgm0tpEM zAyvCO4&Jw!Mc^V(X5>EtAq7VlZ#z$BAq5vJZ##KA8#h}!0x2m1FKS#`R*87FmL&t+cHfg`{i4hOFtR1I0nvK*aa7J_}7o}C$<7KrKoj1(?0 z)Z6s0bn!jmY$#pa+jM>Z>R^3-U^#d4X>_1brvvo&`nd+J>Hsa9I?pwn7f*zp>S{%)Ux2hgFiw|u`3$6H^Q*~n8Z5wOcQ0hLptd*c~j-%msU-k;6Miv8_r0#qjOB9yU zTb&Flrjr+@d?CB39s}PvuQk;_30dQeAt0mUcqC4JGdW&yzST5qO_N9ISsHCMHG%si zomyYL;QLAe+jHO1Mw26&L+@$b!FQ2`7B>t_;$ZV3RS)_eP5GDCoP3znRcim7)63Hu z;dtxpmb&n*vLx3BqJvXxzqYEroqUBl=7e0mYt)pdDySlqYH)2u13y(0i$cCQUxM3E z1b@AivSh1dvLP>XfU4joUDH-+Uxpdg&Y254MMt9;!!uvy`5w8da^JnTu^NLz$FLVW zr6rG~iS6+VwuXMb3g5!Tccdg{#hEBM%MJgsyS#mH;ODkOdv%a517wGXoulp4D*SkP zW&30HnoDHD8m8gqgi!9fy?qRX#BXz=r>WO1wv$~IaPf-gs_@@2P%0nODQ7dvb% zoa?@j3Fb>yH1pq`i`ik5VOJU4>H@NEO+HI>Ch}&Wpv-(1JX%!tMCVz0=@C^y!P891VB`8~&c<5z#%esci zw~u}x?q=od`%aZu);Y#Aj@u@K{^|ia7O%b;VY%}KiE%NzOL8r-khb&s;b7d!34G;1 zzVzuv@IYA3CShaP)er$JArWmcg^2k?f-x@<$JR*k$Bt!d)kV3gP3r3=-Z*C~X>i7v zsioIt{N2l*jg}oA{c&nJZ+z$sj1E6^TlZnfHyf^wGlo9&B@lbM$xBlAOXRe)n}tv= z!er4`HGMy?+CuB5m$q_k)T7iV;p`o5S-D-~^(8d%YrZlrUiXW?hYS*6zMaM*`Yd@D zOSxyA1pP{Ks-eQOSFs0SvLC5@79exB*qFm?;@~*Ro|5B6+B{~k_N3v5>e9gl^p~yk zXV(Y1l?_?a4jCLf|wHynm1X${YF$S?9=y*9A>O@ z{P|g2v8eaOTS~V9_N~IXmB8q{v$FutS z^|PjHec2m|4ByXK{76H}2wDZcsMT+ul^MPH8jZ#%w6Rs9F7v||oA*k#&OJ!Xa`3R; 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