Continuity of Sums and Products 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, multiplication, identities, additive inverses and order of their ordered field structure; 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 $(\mathbb{R},d_{\mathbb{R}})$, and let $c\in\mathbb{R}$. Define maps $f+g$, $fg$ and $cf$ from $A$ to $\mathbb{R}$ by

$$(f+g)(z)=f(z)+g(z),\qquad (fg)(z)=f(z)\cdot g(z),\qquad (cf)(z)=c\cdot f(z).$$

Then the following hold.

1.¶ For every $b\in\mathbb{R}$, the map $A\to\mathbb{R}$ with constant value $b$ is continuous at $x$ relative to $A$.

2.¶ $f+g$ is continuous at $x$ relative to $A$.

3.¶ $fg$ is continuous at $x$ relative to $A$.

4.¶ $cf$ is continuous at $x$ relative to $A$.

5.¶ If $f$ and $g$ are continuous on $A$, then $f+g$, $fg$ and $cf$ are continuous on $A$.