Let $u: \mathbb{R}\to \mathbb{R}$. If $u$ is strongly differentiable (i.e. differentiable in the classic sense) with strong derivative $u'$, then $u$ is also weakly differentiable and every weak derivative equals $u'$ almost everywhere.
Now, assume $u$ is continuous and has a continuous weak derivative: can we conclude that $u$ is continuously differentiable in the strong (i.e. usual) sense?
Best Answer
The weak derivative is unique, so there are no "several derivatives" (notice that we don't work with functions, but rather with class of functions).
For your second question, the answer is yes. This is a simple consequence of the fundamental theorem of calculus. Moreover, instead of saying that "there is a continuous weak derivative", we rather say that "there is a continuous representative of the derivative".