Let $f:(a,b) \to \mathbb R$ be Lipschitz.

The derivative $f'$ exists on some set $D \subset (a,b)$ of full measure and is bounded (by Rademacher).

Is $f'$ continuous (or some representative) on the same set $D$ or on some other set of full measure $C \subset D$?

If not, do you have a counterexample?

Let $f:(a,b) \to \mathbb R$ be Lipschitz.

The derivative $f'$ exists on some set $D \subset (a,b)$ of full measure and is bounded (by Rademacher).

Is $f'$ continuous (or some representative) on the same set $D$ or on some other set of full measure $C \subset D$?
In other words, is $f': C \to \mathbb R$ continuous?

If not, do you have a counterexample?