Derivative zero implies constant

Let $I\subseteq\mathbb{R}$ be an interval and let $f:I\to\mathbb{R}$ be differentiable on $I^\circ$.

Proposition: If $f'(x)=0$ for all $x\in I^\circ$, then $f$ is constant on $I$; i.e., for all $x,y\in I$ one has $f(x)=f(y)$.

This is a fundamental rigidity result: having zero instantaneous rate of change everywhere forces no global change.