Continuity of the Reciprocal of a Nonvanishing Real-Valued Function 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, multiplicative 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 $s\ne t$, write $s^{-1}$ for the multiplicative inverse of $s$ when $s\ne 0$, 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 $f:A\to\mathbb{R}$ satisfy $f(w)\ne 0$ for every $w\in A$, and define $r:A\to\mathbb{R}$ by $r(z)=(f(z))^{-1}$.

Then the following hold.

1. If $f$ is continuous at $x$ relative to $A$ as a map from $(X,d)$ to $(\mathbb{R},d_{\mathbb{R}})$, then so is $r$.

2. If $f$ is continuous on $A$, then so is $r$.