Rolle's Theorem in One Dimension

Let $I$ be an interval in the sense of Interval in the Real Line, let $[a,b]\subseteq I$ with $a<b$, and let $f:I\to\mathbb{R}$ be continuous on $[a,b]$ in the sense of Continuity on a Closed Interval and differentiable at every point of $(a,b)$ in the sense of Derivative at an Interior Point. Assume that $f(a)=f(b)$. Then there exists $c\in(a,b)$ such that