A Continuous Function with Vanishing Derivative is Constant

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 $d_{\mathbb{R}}$. Write $(a,b)$ for $\{x\in\mathbb{R}:a<x<b\}$; every $x\in(a,b)$ is an interior point of the interval $[a,b]$, since $a,b\in[a,b]$ and $a<x<b$.

Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and differentiable at every point $x\in(a,b)$, with

$$f'(x)=0\qquad\text{for every }x\in(a,b) .$$

Then $f$ is constant:

$$f(x)=f(a)\qquad\text{for every }x\in[a,b] .$$