Uniqueness of the Derivative at an Interior Point

Let $\mathbb{R}$ be the real numbers, let $I\subseteq\mathbb{R}$ be order-convex, let $f:I\to\mathbb{R}$, and let $x_0\in I$ satisfy $u<x_0<v$ for some $u,v\in I$, so that $x_0$ is an interior point of $I$.

Suppose that $L$ and $L'$ are real numbers each of which has the property required of the derivative in Derivative at an Interior Point, that is, for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there exists $\delta\in\mathbb{R}$ with $0<\delta$ such that every $h\in\mathbb{R}$ with $0<|h|<\delta$ and $x_0+h\in I$ satisfies

and likewise for $L'$, where $|\,\cdot\,|$ is the absolute value.

Then $L=L'$. In particular the derivative $f'(x_0)$ is well defined.