The Square of a Nonnegative Continuous Real-Valued Function is Continuous

Let $(X,d)$ be a metric space, let $A\subseteq X$, and let $\mathbb{R}$ be the set of real numbers with the addition, multiplication and order $\le$ of its ordered field structure, regarded as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line.

Let $f:A\to\mathbb{R}$ be continuous on $A$ relative to $A$, as a map into $(\mathbb{R},d_{\mathbb{R}})$, and suppose that $0\le f(y)$ for every $y\in A$. Let $f^{2}:A\to\mathbb{R}$ be the function with $f^{2}(y)=f(y)\,f(y)$ for $y\in A$.

Then $f^{2}$ is continuous on $A$ relative to $A$.