Hello,

While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated.

QUESTION

Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $f\in L^1_{\text{loc}}(U)$ such that

1. the classical derivative $Df$ exists everywhere in $U$.

2. $f$ is weakly differentiable in $U$. Let us write $D_w f$ to denote the weak derivative of $f$.

3. $Df\neq D_w f$, on a set of positive measure.

Note that, we are assuming the existence of both the derivatives. I'm aware of examples where one exists while other one does not.

The problem seems to be related to the question of validity of integration by parts for differentiable functions.

Thank you.