Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space

Let $(X,d)$ be a metric space, let $A\subseteq X$, and let $x\in A$. Let $\mathbb{R}$ denote the real numbers, with the addition, identities, additive inverses and order of their ordered field structure, the order $\le$ being in particular a total order; for $s,t\in\mathbb{R}$ write $s-t$ for $s+(-t)$, write $s<t$ to mean that $s\le t$ and $s\ne t$, let $|s|$ denote the absolute value of $s$, and let $d_{\mathbb{R}}$ be given by $d_{\mathbb{R}}(s,t)=|s-t|$, which is a metric on $\mathbb{R}$ by The Absolute Value Metric on the Real Line.

Let $f,g:A\to\mathbb{R}$ be continuous at $x$ relative to $A$ as maps from $(X,d)$ to $(\mathbb{R},d_{\mathbb{R}})$. Define maps from $A$ to $\mathbb{R}$ by

$$|f|(z)=|f(z)|,\qquad \max\{f,g\}(z)=\max\{f(z),g(z)\},\qquad \min\{f,g\}(z)=\min\{f(z),g(z)\},$$

where $\max$ and $\min$ of two elements are as in the definition of the maximum and the definition of the minimum.

Then the following hold.

1. $|f|$ is continuous at $x$ relative to $A$.

2. $\max\{f,g\}$ is continuous at $x$ relative to $A$.

3. $\min\{f,g\}$ is continuous at $x$ relative to $A$.

4. If $f$ and $g$ are continuous on $A$, then $|f|$, $\max\{f,g\}$ and $\min\{f,g\}$ are continuous on $A$.