Weak and classical derivatives: an overview

I am studying PDE's and we have defined the following notion of weak derivative:

Given a domain $$\Omega\subset\mathbb{R}^{n}$$ a function $$f\in L^1_{loc}(\Omega)$$ is wealy differentiable with respect to the $$i-th$$ variable if there exist a function $$f_{x_i}\in L^1_{loc}$$ such that for all test functions $$\phi\in C^{\infty}_c(\Omega)$$ we have $$\int f_{x_i}\phi =-\int f\phi_{x_i}$$

Now my questions are:
When the weak and classical and weak derivative do or do not coincide?
I can think that they coincide $$f\in C^1$$, while they cannot coincide when f is piecewise $$C^1$$ and shows jumps (since then we get a Dirac $$\delta$$ in the distributional derivative).
If the classical derivative presents a singularity whcich is not a jump we have to test its action on the generic $$\phi$$ (this boils down to showing the singularity is summable).
Is the definition of weak derivative the most safe and convenient way to compute it ?

Thanks for help!

In $$\mathbb R$$:
• If $$f$$ is derivable everywhere, then weak derivative and derivative coincides.
• If $$f$$ is absolutely continuous, then weak derivative and strong derivative coincides (although strong derivative is defined only a.e.)
• If $$f$$ is a.e. derivable (but not absolutely continuous), then the weak derivative may not exist or be different (example : Cantor function $$F$$ is derivable a.e. it's derivative if $$0$$, but if the weak derivative exist they can't coincides because otherwise $$\forall \varphi \in \mathcal C_0^\infty [0,1],\int F\varphi '=0\implies \exists C>0: F=C\text{ a.e.}$$ which doesn't hold.)