Characterization of globally lipschitz function in terms of derivatives

Question

I would like to know if the following generalization of the characterization of globally lipschitz functions in the class $\mathcal{C}^1$:

Let $f:I \to \mathbb{R}$, $I \subseteq \mathbb{R}$ an interval.

If $f \in \mathcal{C}^1$ then $f$ is globally lipschitz $\iff \exists L \ge 0.\forall t \in I.|f'(t)| \le L$

The generalization goes as follows:

A scalar function is globally lipschitz iff its derivative exists and is bounded except in a set of measure zero.

Is this still true? Has it been studied/asked before?

Answer 1

It is not true. The Cantor-Vitali function is continuous in $[0,1]$, differentiable almost everywhere with zero derivative, but it is not Lipschitz continuous.