Differentiability at an Interior Point is a Local Property

In the setting of The Real Numbers: Standing Notation and Background, let $I$ and $J$ be intervals in $\mathbb{R}$ with $I\subseteq J$, let $f:J\to\mathbb{R}$, and let $f|_{I}$ denote the restriction of $f$ to $I$. Let $x_{0}$ be a point of $I$ that is an interior point of $I$ and also an interior point of $J$, and let $|t|$ denote the absolute value of $t\in\mathbb{R}$. Suppose that there is a real number $r>0$ such that

¶ If $f|_{I}$ is differentiable at $x_{0}$ with derivative $L$, then $f$ is differentiable at $x_{0}$ with derivative $L$.