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Suppose $BV$ functions, where the notion of approximate gradient is importantf \in W^{1,1}_{loc}(U)$. http://www.encyclopediaofmath.org/index.php/Approximate_differentiability The approximate gradient is the measure theoretic equivalent of the differentialThen no, defined as the object since for such an$f$, we have that$Df$exists and the difference quotients are small on a set of full or large measureapproximate limit$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$. In particular, every Sobolev function is approximately differentiable and the weak derivative classical differential is a candidate for the approximate differential, unique up to a set of Lebesgue measure zero. But the strong derivative is also a candidate, and so they coincide$Df=\nabla f$wherever the two exist, and hence in$U$up to a set of Lebesgue measure zero. We therefore overcome the difficulty of not assuming$f \in C^1(U)$or Lipschitz by using a definition which is not for integrable functions (approximate differentiability). http://www.encyclopediaofmath.org/index.php/Approximate_differentiability 1 I felt the answer was no from the time I read your question, and now I think I have an argument to give you. This comes from the context of studying$BV$functions, where the notion of approximate gradient is important. http://www.encyclopediaofmath.org/index.php/Approximate_differentiability The approximate gradient is the measure theoretic equivalent of the differential, defined as the object such that the difference quotients are small on a set of full or large measure. In particular, every Sobolev function is approximately differentiable and the weak derivative is a candidate for the approximate differential, unique up to a set of Lebesgue measure zero. But the strong derivative is also a candidate, and so they coincide, up to a set of Lebesgue measure zero. We therefore overcome the difficulty of not assuming$f \in C^1(U)\$ or Lipschitz by using a definition which is not for integrable functions (approximate differentiability).