A.e. differentiable weak derivative

Question

I’m studying Sobolev Spaces. I’m trying to understand if exists a function with an a.e differentiable weak derivative. Could anyone show me an example (if there exists) please?

Answer 1

It's a general fact that if $a<b,$ then $H^1([a,b])$ consists of functions of the form $f(x)=C+\int_a^x g(t)\textrm{d}t$ where $g\in L^2([a,b])$ and in that case, $g=f'$ in the weak sense. Now, $L^2([a,b])\subseteq L^1([a,b]),$ so $f$ will be continuous, and furthermore, see for instance Evans' book on Partial Differential Equations, $f$ will be differentiable almost everywhere with ordinary derivative $g$.

Hence, take your favourite discontinuous $L^2$-function $h$ and define $f(x)=\int_a^x\int_a^s h(s)\textrm{d}s\textrm{d}t$. Then, $f'(x)=g(x):=\int_a^x h(s)\textrm{d}s$ in the weak sense, which is a.e. differentiable. Both $f$ and $f'$ are $L^2$ functions, since they are continuous and hence, bounded.