Proof of Differentiability at an Interior Point is a Local Property

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named in the step where it is cited.

By Derivative at an Interior Point, the hypothesis on $f|_{I}$ says: for every real $\varepsilon>0$ there is a real $\delta>0$ such that every $h\in\mathbb{R}$ with $0<|h|<\delta$ and $x_{0}+h\in I$ satisfies

What is to be proved is the corresponding statement for $f$ on $J$: that for every real $\varepsilon>0$ there is a real $\delta'>0$ such that every $h\in\mathbb{R}$ with $0<|h|<\delta'$ and $x_{0}+h\in J$ satisfies

This is meaningful because $x_{0}$ is by hypothesis an interior point of $J$.

Let $\varepsilon$ be a real number with $0<\varepsilon$. The choices are made in this order: $\varepsilon$ is given, then $\delta$, then $\delta'$. Choose $\delta>0$ as in the hypothesis, and let $\delta'$ be the least of the two numbers $\delta$ and $r$, so that $\delta'\le\delta$, $\delta'\le r$ and $\delta'$ is one of $\delta$ and $r$, by claim 9 of Elementary Order Arithmetic in an Ordered Field; since both $\delta$ and $r$ are positive, so is $\delta'$.

Let $h\in\mathbb{R}$ satisfy $0<|h|<\delta'$ and $x_{0}+h\in J$, and put $y=x_{0}+h$. Then

so $y$ is a point of $J$ with $|y-x_{0}|<r$, and the hypothesis on $r$ places $y$ in $I$. Consequently $f|_{I}(y)=f(y)$, and $f|_{I}(x_{0})=f(x_{0})$ because $x_{0}\in I$. Moreover $0<|h|<\delta'\le\delta$ and $x_{0}+h=y\in I$, so the displayed bound of the hypothesis applies to this $h$ and yields

As $\varepsilon>0$ was arbitrary, the real number $L$ satisfies the condition of Derivative at an Interior Point for $f$ at the interior point $x_{0}$ of $J$. Hence $f$ is differentiable at $x_{0}$ with derivative $L$, which is the assertion.