A Lipschitz Function on an Open Interval is Differentiable Almost Everywhere

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background 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 Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, 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 $\mathbb{R}$; null has the meaning fixed in that setting, and Lipschitz is understood for the Euclidean distances.

Let $I\subseteq\mathbb{R}$ be a nonempty open interval, let $L\in\mathbb{R}$ with $0\le L$, and let $g:I\to\mathbb{R}$ be Lipschitz with constant $L$, that is

$$|g(x)-g(y)|\le L\,|x-y|\qquad\text{for all }x,y\in I .$$

Let $D$ be the set of those $x\in I$ at which $g$ has a derivative $g'(x)$, a real number. Then the following hold.

1. (Almost everywhere differentiability) (i) The set $I\setminus D$ is null.

2. (Bound on the derivative) (ii) For every $x\in D$ one has $|g'(x)|\le L$.