We know that a function $f(x)$ is locally integrable over a finite region $\Omega$ if: $$\int_{\Omega}|f(x)|\,dx < \infty$$ Some non-differentiable functions fulfill that criteria, for example: $|x|$, $e^{|x|}$, or the step functions.
We also know that every locally integrable function f(x) can create a distribution $F[\phi]$, with $\phi(x)$ being a bump function: $$F[\phi] = \int_{-\infty}^{\infty}f(x)\phi(x)dx$$ Finally we also know the definition of a distributional derivative: $$F'[\phi] = \int_{-\infty}^{\infty}f'(x)\phi(x)dx$$ But how does one interpret $f'(x)$ if $f$ is not necessarily differentiable?
Let's say $f(x)=|x|$. Even if I use some questionable math and assume $f'(x) = $ sign$(x)$, I still couldn't evaluate the integral because the Fundamental Theorem of Calculus only holds for continuous antiderivatives. So, is $f'(x)$ suggesting that I should split up the integral? $$(|x|)'[\phi] = \int_{-\infty}^{\infty}(|x|)'\phi(x)dx=\int_{-\infty}^{0}(-x)'\phi(x)dx+\int_{0}^{\infty}(x)'\phi(x)dx$$
Or is the distributional derivative only defined if $f(x)$ is differentiable?