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 functions that are only differentiable.

Thank you.

a priorionly defined as a distribution, which among other things means that hypothesis (3) is not well-formed without an additional hypothesis on either the weak derivative (to interpret it as a function or measure) or the strong derivative (to interpret it as a measure or distribution). ... – Terry Tao Dec 17 '12 at 20:41