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\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \)
Math and science::Analysis::Tao::09. Continuous functions on R

Arithmetic preserves continuity

Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) and \( g : X \to \mathbb{R} \) be functions, and let \( x_0 \) be an element of \( X \). Then, if \( f \) and \( g \) are continuous at \( x_0 \), then so too are the functions [...], [...], [...], [...] and [...]. If \( g \) is non-zero on \( X \), then [...] is also continuous at \( x_0 \).