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principali della continuità e dei limiti di funzione/main.tex b/Analisi matematica 1/Parte teorica/2023-03-23, 24, 28, Proprietà principali della continuità e dei limiti di funzione/main.tex index 38d19a4..cad1b44 100644 --- a/Analisi matematica 1/Parte teorica/2023-03-23, 24, 28, Proprietà principali della continuità e dei limiti di funzione/main.tex +++ b/Analisi matematica 1/Parte teorica/2023-03-23, 24, 28, Proprietà principali della continuità e dei limiti di funzione/main.tex @@ -32,28 +32,28 @@ \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: + costruisce allora la successione $(z_n)_{n \in \NN} \subseteq X \setminus \{\xbar\}$ nel seguente modo: - \[ z_n = \system{x_{\frac{n}{2}} & \text{se } n \text{ è pari}, \\ y_{\frac{n-1}{2}} & \text{altrimenti},} \] + \[ z_n = \system{x_n & \se n \epari, \\ y_n & \altrimenti,} \] \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 $y_n$. \\ + elementi di $x_{2n}$ e a quelli dispari quelli di $y_{2n+1}$. \\ - 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 $n \geq 2 n_k$, + Si mostra che $z_n \tendston \xbar$. Sia $I$ un intorno di $\xbar$. Allora, dal momento che le due sottosuccessioni + $(x_{2n}), (y_{2n+1}) \tendston \xbar$, esistono sicuramente due + $n_x, n_y \in \NN$ tali che $n \geq n_x \implies x_{2n} \in I$ e $n \geq n_y \implies y_{2n+1} \in I$. Pertanto, + detto $n_k = \max\{n_x, n_y\}$, $n \geq n_k \implies x_{2n}, y_{2n+1} \in I$, 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 + $f(x_{2n})$ e $f(y_{2n+1})$ convergono per ipotesi 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 + Data $(x_n) \subseteq \RR$, sia $f : \NN \to \RRbar$ tale che $f(n) := x_n$, $\forall n \in \NN$. Allora $f(n) \tendston L \iff x_n \tendston L$. \end{proposition} @@ -63,9 +63,9 @@ 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 $m$. + si consideri il minimo di tale insieme, occorre semplicemente che $A$ sia non vuoto e che sia infinito.}, 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 $x_n \tendston L$. \\ + $f(n) \in I$, $\forall n \geq m$, e quindi che $x_n \in I$, 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 @@ -84,7 +84,7 @@ \begin{proof} 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 + entrambe le ipotesi che esiste un intorno $J$ di $\xbar$ tale che $f(J \cap X \setminus \{\xbar\}) \subseteq I$, e quindi, per definizione, la tesi. \end{proof} @@ -93,8 +93,7 @@ in $\xbar$. Pertanto per rendere la proposizione precedente vera, è necessario ipotizzare che $\xbar$ sia un punto di accumulazione (infatti il limite in un punto isolato - non esiste per definizione, mentre in tale punto $f$ è - continua). + non esiste per definizione). \end{remark} \begin{proposition} @@ -129,7 +128,9 @@ \begin{remark} 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. + di definizione\footnote{Tale fatto è una mera conseguenza della + derivabilità delle funzioni elementari nel proprio insieme di + definizione.}. \end{remark} \begin{proposition} @@ -145,10 +146,10 @@ 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 - $f(x) \in J \cap Y \setminus \{f(\xbar)\} \implies g(f(x)) \in g(J \cap Y \setminus \{f(\xbar)\}) \subseteq I$. + $f(x) \in J \cap Y \setminus \{f(\xbar)\} \implies g(f(x)) \in g(J \cap Y \setminus \{f(\xbar)\}) \subseteq I$, da cui la tesi. \end{proof} - \begin{theorem} + \begin{theorem} (di sostituzione nel limite) 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 $Y$ e $g : Y \to \RRbar$ @@ -196,9 +197,29 @@ \end{proof} \begin{exercise} - Mostrare che tutte le ipotesi della proposizione precedente sono necessarie, fornendo alcuni controesempi. + Si mostri che tutte le ipotesi della proposizione precedente sono necessarie, fornendo un controesempio. \end{exercise} + \begin{solution} + Chiaramente $\xbar$ e $\ybar$ devono essere punti di + accumulazione dei propri insiemi di appartenenza, altrimenti + non sarebbe possibile calcolarne il limite. \\ + + Inoltre, se $\ybar \in Y$ è necessario che $g$ sia anche continua in + $\ybar$ (nella dimostrazione della proposizione si è infatti + utilizzato il fatto che $g(\ybar) = \zbar$). Se così non dovesse + essere, si potrebbero definire + due funzioni $f$ e $g$ in modo tale che: + + \[ f(x) = 0, \qquad g(y) = \system{ 0 & \se y = 0, \\ 1 & \altrimenti.} \] + + \vskip 0.05in + + Si osserva subito che $g(y)$ non è continua in $0$. Inoltre + $f(x) \tendsto{0} 0$ e $g(y) \tends{y \to 0} 1$. Tuttavia + $g(f(x)) = g(0) = 0 \tendsto{0} 0 \neq 1$, da cui il controesempio. + \end{solution} + \begin{proposition} Date $f_1, f_2 : X \to \RR$ continue in $\xbar$. Allora: @@ -214,23 +235,25 @@ \begin{enumerate}[(i)] \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 $\abs{f(x) - f(\xbar)} \leq + \implies \abs{f_1(x) - f_1(\xbar)}$, $\abs{f_2(x) - f_2(\xbar)} \leq \eps$ (per ogni $\eps > 0$, è infatti sufficiente considerare $\delta = \min\{\delta_1, \delta_2\}$, ossia il minimo delle semilunghezze degli intorni + di $\xbar$ rispetto a $f_1$ ed $f_2$). Allora, per la + disuguaglianza triangolare, $\abs{f(x) - f(\xbar)} \leq \abs{f_1(x) - f_1(\xbar)} + \abs{f_2(x) - f_2(\xbar)} \leq 2\eps$. - Si conclude dunque che $\forall \eps > 0$, $\exists \delta > 0 + Si ricava 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 $\xbar$. + $2\eps \tends{\eps \to 0^+} 0$, si conclude anche 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) + $\forall \eps > 0$, $\exists \delta > 0$ tale che $\abs{x - \xbar} < \delta \implies \abs{f_1(x) - f_1(\xbar)}$, $\abs{f_2(x) - f_2(\xbar)} < \eps$ (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 $\abs{e_1}, \abs{e_2} < \eps$. + precedente). Allora $f_1(x) = f_1(\xbar) + e_1$ e + $f_2(x) = f_2(\xbar) + e_2$, con $\abs{e_1}, \abs{e_2} < \eps$ + e $\abs{x - \xbar} < \delta$. 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 + \eps \abs{f_1(\xbar)} + \eps^2}_{\eps'}$. Poiché $\eps' \tends{\eps \to 0^+} 0$, si ricava che $\forall \eps > 0$, $\exists \delta > 0 \mid \abs{x - \xbar} < \delta \implies \abs{f_1(x) f_2(x) - f_1(\xbar) f_2(\xbar)} = \abs{e} \leq \eps'$, ossia si conclude che $f_1 f_2$ è continua in $\xbar$. \end{enumerate} \end{proof} @@ -259,38 +282,38 @@ f_2(x) & \text{altrimenti}. \end{cases} \] - \vskip 0.05in + \vskip 0.1in - Si dimostrano allora i due risultati separatamente. \\ + Si dimostrano ora i due risultati separatamente. \\ \begin{enumerate}[(i)] \item Si definisce $\widetilde{f_1 + f_2} : X \cup \{\xbar\} \to \RRbar$ nel seguente modo: \[ \widetilde{f_1 + f_2}(x) = \system{L_1 + L_2 & \text{se } x = \xbar, \\ f_1(x) + f_2(x) & \text{altrimenti}.} \] - La somma $L_1 + L_2$ è ben definita dacché sia $L_1$ che $L_2$ sono elementi di $\RR$. + Si osserva che la somma $L_1 + L_2$ è ben definita dacché sia $L_1$ che $L_2$ sono elementi di $\RR$. Poiché da una proposizione precedente $\tilde{f_1}$ e $\tilde{f_2}$ sono continue in $\xbar$, $\tilde{f_1} + \tilde{f_2}$ è continua anch'essa in $\xbar$. È sufficiente allora dimostrare che $\widetilde{f_1 + f_2} = \tilde{f_1} + \tilde{f_2}$. Se $x \neq \xbar$, $\widetilde{f_1 + f_2}(x) = f_1(x) + f_2(x) = \tilde{f_1}(x) + \tilde{f_2}(x) = (\tilde{f_1} + \tilde{f_2})(x)$. Se invece $x = \xbar$, $\widetilde{f_1 + f_2}(x) = L_1 + L_2 = \tilde{f_1}(x) + \tilde{f_2}(x) = (\tilde{f_1} + \tilde{f_2})(x)$. Quindi $\widetilde{f_1 + f_2} = - \tilde{f_1} + \tilde{f_2}$, e si conclude che $\widetilde{f_1 + f_2}$ è dunque continua in $\xbar$, ossia + \tilde{f_1} + \tilde{f_2}$, e dunque si conclude che $\widetilde{f_1 + f_2}$ è continua in $\xbar$, ossia che $(f_1 + f_2)(x) = f_1(x) + f_2(x) \tendsto{\xbar} L_1 + L_2$. \item Si definisce, analogamente a prima, $\widetilde{f_1 f_2} : X \cup \{\xbar\} \to \RRbar$ nel seguente modo: \[ \widetilde{f_1 f_2}(x) = \system{L_1 L_2 & \text{se } x = \xbar, \\ f_1(x) f_2(x) & \text{altrimenti}.} \] - Il prodotto $L_1 L_2$ è ben definito dacché sia $L_1$ che $L_2$ sono elementi di $\RR$. + Come prima, si osserva che il prodotto $L_1 L_2$ è ben definito dacché sia $L_1$ che $L_2$ sono elementi di $\RR$. Poiché da una proposizione precedente $\tilde{f_1}$ e $\tilde{f_2}$ sono continue in $\xbar$, $\tilde{f_1} \tilde{f_2}$ è continua anch'essa in $\xbar$. È sufficiente allora dimostrare che $\widetilde{f_1 f_2} = \tilde{f_1}\tilde{f_2}$. Se $x \neq \xbar$, $\widetilde{f_1 f_2}(x) = f_1(x) f_2(x) = \tilde{f_1}(x) \tilde{f_2}(x) = (\tilde{f_1}\tilde{f_2})(x)$. Se invece $x = \xbar$, $\widetilde{f_1 f_2}(x) = L_1 L_2 = \tilde{f_1}(x) \tilde{f_2}(x) = (\tilde{f_1} \tilde{f_2})(x)$. Quindi $\widetilde{f_1 f_2} = - \tilde{f_1} \tilde{f_2}$, e si conclude che $\widetilde{f_1 f_2}$ è dunque continua in $\xbar$, ossia + \tilde{f_1} \tilde{f_2}$, da cui si conclude che $\widetilde{f_1 f_2}$ è anch'essa continua in $\xbar$, ossia che $(f_1 f_2)(x) = f_1(x) f_2(x) \tendsto{\xbar} L_1 L_2$. \end{enumerate} \end{proof} \begin{definition} (intorno destro e sinistro) Se $\xbar \in \RR$, si dicono - \textbf{intorni destri} gli intervalli della forma $[\xbar, \xbar + \eps]$ con + \textbf{intorni destri} di $\xbar$ 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} @@ -304,17 +327,20 @@ \begin{definition} (limite destro e sinistro) Sia $\xbar$ un punto di accumulazione - destro di $X$. Allora $\lim_{x \to \xbar^+} f(x) = L \defiff \forall I$ + destro di $X$. Allora si dice che $f$ ammette un \textbf{limite destro} $L$ in $\xbar$, $\lim_{x \to \xbar^+} f(x) = L$, se e solo se $\forall I$ 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. + il \textbf{limite sinistro}: $\lim_{x \to \xbar^-} f(x) = L \defiff \forall I$ + intorno di $L$, $\exists J$ intorno sinistro di $\xbar$ tale che + $f(J \cap X \setminus \{\xbar\}) \subseteq I$. \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 $f$. + (continuità destra e sinistra) Sia $\xbar \in X$. Allora $f$ è \textbf{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 dice che $f$ è \textbf{continua a sinistra} su $\xbar$ se e solo se $\forall I$ intorno + di $f(\xbar)$, $\exists J$ intorno sinistro di $\xbar$ tale che + $f(J \cap X \setminus \{\xbar\}) \subseteq I$. \end{definition} \begin{remark} @@ -334,25 +360,29 @@ \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 + Analogamente esiste il limite sinistro se $\xbar$ è invece un punto di accumulazione sinistro di $X$. \end{proposition} \begin{proof} Senza perdità di generalità, si assuma $f$ crescente (per il caso decrescente è sufficiente considerare - $g(x) = -f(x)$) Si consideri l'insieme: + $g = -f$). Si consideri allora l'insieme: - \[E = \{ f(x) \mid x > \xbar \text{ e } x \in X \}.\] + \[E = \{ f(x) \mid x > \xbar \E x \in X \}.\] + + \vskip 0.05in Si consideri adesso $L = \inf E$ e un suo intorno $I$. Se non - esistesse un intorno destro $J$ di $\xbar$ tale che $f(J \cap X \setminus \{\xbar\}) \subseteq I$, allora - $\sup I$ sarebbe un minorante di $E$ maggiore di $L$, \Lightning. Quindi tale $J$ esiste, da cui la tesi. - Analogamente per il caso sinistro. + esistesse $x > \xbar$ tale che $f(x) \in I$, $\sup I$ sarebbe + un minorante di $E$ maggiore\footnote{Infatti $f(x) \geq L$ dacché è $L$ è un minorante di $E$, da cui $f(x) \notin I \implies + f(x) > \sup I$.} di $L$, \Lightning. Quindi esiste + $x > \xbar \mid f(x) \in I$, e dal momento che $f$ è + crescente, l'intorno destro $J$ di $\xbar$ di raggio $x - \xbar$ sarebbe + tale che $f(J \cap X \setminus \{\xbar\}) \subseteq I$, da cui la tesi. \end{proof} - %TODO: migliorare dimostrazione - - \begin{example} (funzione discontinua in ogni punto di $\RR$) Si consideri la funzione $f : \RR \to \RR$ definita + \begin{example} (funzione discontinua in ogni punto di $\RR$) Si consideri la funzione\footnote{Tale funzione è detta \textit{funzione di Dirichlet}, + in onore al matematico tedesco Peter Dirichlet (1805 -- 1859).} $f : \RR \to \RR$ definita nel seguente modo: \[ f(x) = \system{ 1 & \text{se }x \in \QQ, \\ 0 & \text{altrimenti}, } \] @@ -363,13 +393,13 @@ in nessun punto di $\RR$. Sia infatti $\xbar \in \RR \setminus \QQ$. Dal momento che $\QQ$ è denso in $\RR$, $\xbar$ è un punto di accumulazione di $\QQ$, e quindi esiste una successione $(x_n) \subseteq \QQ$ tale che $x_n \tendston \xbar$. Se $f$ fosse continua in $\xbar$, $\lim_{n \to \infty} f(x_n) = 0$, - ma per l'intorno $I = [-\frac{1}{2}, \frac{1}{2}]$ non esiste alcun $n_k$ tale per cui $f(x_n) \in I$ $\forall n + ma per l'intorno $I = [0-\frac{1}{2}, 0+\frac{1}{2}]$ di $0$ non esiste alcun $n_k$ tale per cui $f(x_n) \in I$ $\forall n \geq n_k$, dal momento che, per definizione di $f$, $f(x_n) = 1$ $\forall n \in \NN$. Quindi $f$ non è continua in nessun $\xbar \in \RR \setminus \QQ$. \\ Sia ora $\xbar \in \QQ$. $\xbar$ è un punto di accumulazione di $\RR \setminus \QQ$ (si può infatti considerare la successione $(x_n) \subseteq \RR \setminus \QQ$ definita da $x_n = \xbar + \frac{\sqrt{2}}{n}$, - che è tale che $x_n \tendston \xbar$). Analogamente a come visto prima, allora, per l'intorno $I = [\frac{1}{2}, \frac{3}{2}]$, $f(x_n) \notin I$ $\forall n \in \NN$, e quindi $f$ non è continua neanche su $\xbar \in \QQ$. + che è tale che $x_n \tendston \xbar$). Analogamente a come visto prima, allora, per l'intorno $I = [1-\frac{1}{2}, 1+\frac{1}{2}]$ di $1$, $f(x_n) \notin I$ $\forall n \in \NN$, e quindi $f$ non è continua neanche su $\xbar \in \QQ$, ossia è discontinua ovunque. \end{example} \begin{exercise} @@ -378,7 +408,7 @@ \end{exercise} \begin{solution} - Si assuma $f$ crescente, senza perdita di generalità (altrimenti è sufficiente considerare $g(x) = -f(x)$). + Si assuma $f$ crescente, senza perdita di generalità (altrimenti è sufficiente considerare $g = -f$). Sia $E$ l'insieme dei punti di discontinuità di $f$. $\forall \xbar \in E$, $\xbar$ è un punto di accumulazione destro e sinistro di $I$ (infatti $I$ è un intervallo), ed in particolare esistono sempre il limite destro $L^-(\xbar)$ ed il limite sinistro $L^+(\xbar)$ in $\xbar$ (dal momento che $f$ è monotona), e sono tali che $L^+(\xbar) > L^-(\xbar)$ (sicuramente diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index ca0a299..0bba8d0 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -27,10 +27,11 @@ \let\oldvec\vec \renewcommand{\vec}[1]{\underline{#1}} -\newcommand{\altrimenti}{\text{altrimenti }} +\newcommand{\E}{\text{ e }} +\newcommand{\altrimenti}{\text{altrimenti}} \newcommand{\se}{\text{se }} -\newcommand{\epari}{\text{è dispari}} -\newcommand{\edispari}{\text{è pari}} +\newcommand{\epari}{\text{ è pari}} +\newcommand{\edispari}{\text{ è dispari}} \newcommand{\nl}{\ \\}