Extreme Value Theorem on a Closed Interval

Let $\mathbb{R}$ be the real numbers and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, that is, $\mathbb{R}$ equipped with the absolute value metric. Let $a,b\in\mathbb{R}$ satisfy $a\le b$, and let $[a,b]$ be the closed interval with endpoints $a$ and $b$.

Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$, as a map from the subset $[a,b]$ of $(\mathbb{R},d_{\mathbb{R}})$ into $(\mathbb{R},d_{\mathbb{R}})$.

Then there exist points $x_{\min},x_{\max}\in[a,b]$ such that