Rolle's Theorem on an Open Interval

Let $\mathbb{R}$ be the set of real numbers, with its order $\le$ and the associated strict order $<$.

Let $p,q\in\mathbb{R}$, let $g:(p,q)\to\mathbb{R}$ be differentiable at every point of the open interval $(p,q)$, and let $a,b\in(p,q)$ satisfy $a<b$ and $g(a)=g(b)$.

Then there exists $c\in(a,b)$ such that