Rolle's Theorem in One Dimension

Let $a,b\in \mathbb{R}$ with $a<b$, and let $f:[a,b]\to\mathbb{R}$ be continuous at every point in $[a,b]$ and differentiable at every point in $(a,b)$. Assume that $f(a)=f(b)$. Then there exists $c\in(a,b)$ such that