Extreme Value Theorem on a Closed Real Interval

Let $a,b$ be real numbers with $a\le b$ in the order of the ordered field $\mathbb{R}$, let $[a,b]$ be the closed interval determined by $a$ and $b$, regarded as a subset of the real line $(\mathbb{R},d_{\mathbb{R}})$, and let the codomain $\mathbb{R}$ carry the same metric $d_{\mathbb{R}}$. Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$.

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

$$f(x_{\min})\le f(x)\le f(x_{\max})\qquad\text{for every }x\in[a,b] .$$