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.



Classical Derivative, Weak Derivative and Integration by Parts
Suppose $f \in W^{1,1}\_{loc}(U)$. Then no, since for such an $f$, we have that $Df$ exists and the approximate limit


$ap\lim\_{y\to x} \frac{f(x)-f(y)-Df(x)(x-y)}{|x-y|} = 0$


exists for almost every $x$, while from assuming classical differentiability we have


$\lim\_{y\to x} \frac{f(x)-f(y)-\nabla f(x)(x-y)}{|x-y|} = 0$


exists for every $x \in U$. In particular, the classical differential is a candidate for the approximate differential, and so $Df=\nabla f$ wherever the two exist, and hence in $U$ up to a set of measure zero.


<http://www.encyclopediaofmath.org/index.php/Approximate_differentiability>


