Let $X$ be a subset of $\mathbf{R}$, and let $f: X \to \mathbf{R}$ and $g : X \to R$ be functions. Let $x_0 \in X$. Then if $f$ and $g$ are both continuous at $x_0$, then the functions $f + g$, $f - g$, $\operatorname{max}(f, g)$, $\operatorname{min}(f, g)$ and $f g$ are also continuous at $x_0$. If $g$ is non-zero on $X$, then $f/g$ is also continuous at $x_0$.

Homepage › Solution manuals › Terence Tao › Analysis I › Proposition 9.4.9 (Arithmetic preserves continuity)

Proposition 9.4.9 (Arithmetic preserves continuity)

Let $X$ be a subset of $R$ , and let $f : X \to R$ and $g : X \to R$ be functions. Let $x_{0} \in X$ . Then if $f$ and $g$ are both continuous at $x_{0}$ , then the functions $f + g$ , $f - g$ , $\max (f, g)$ , $\min (f, g)$ and $fg$ are also continuous at $x_{0}$ . If $g$ is non-zero on $X$ , then $f ∕ g$ is also continuous at $x_{0}$ .

Proposition 9.4.9 (Arithmetic preserves continuity)

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