If a function is a.e differentiable on a bounded open set, is the weak gradient automatically equal to the ordinary gradient?

Question

Let $U$ be a bounded open set in $\mathbb{R}^n$ and $f$ be a real-valued a.e. differentiable function on $U$.

Then, I think, by the definition of the weak derivative, $f$ of course has a weak gradient and this weak gradient equals a.e the ordinary gradient of $f$.

Is my judgement right?

Answer 1

No. The Cantor step function has derivative zero almost-everywhere, but it does not satisfy the definition of weak gradient.