A Nondecreasing Function on an Open Interval is Differentiable Almost Everywhere

We work in the setting of Euclidean Space and Lebesgue Measure: Standing Notation with the dimension $1$, throughout identifying a point of $\mathbb{R}^{1}$ with its single coordinate, so that $\mathbb{R}^{1}$ and $\mathbb{R}$ are written interchangeably; under this convention the Euclidean norm of a point is its absolute value and $d_{E}(s,t)=|s-t|$, by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so that the closed ball $\bar{B}(y,r)$ is the closed interval with endpoints $y-r$ and $y+r$. By Lebesgue Measure on $\mathbb{R}^n$ the measure $\lambda_{1}$ is the Lebesgue measure $\lambda$ on the Borel $\sigma$-algebra of $\mathbb{R}$. Accordingly $\lambda^{\ast}$ denotes Lebesgue outer measure on subsets of $\mathbb{R}$, and null has the meaning fixed in that setting.

Let $I\subseteq\mathbb{R}$ be a nonempty open interval and let $g:I\to\mathbb{R}$ be nondecreasing on $I$, that is $g(x)\le g(y)$ whenever $x,y\in I$ and $x\le y$. Let $D$ be the set of those $x\in I$ at which $g$ has a derivative $g'(x)$, a real number.

Then $I\setminus D$ is null. ¶