In general, what can be said about differentiability of (real-valued) regulated functions, i.e. such for which the left and right limit exist at every point?
Such functions are necessarily continuous except at countably many points, but "how many" points of differentiability do they have? Are they differentiable almost everywhere? Is there a nowhere differentiable regulated function?
A quick google search doesn't seem to help..
Yes, there is. Even more: there are continuous functions that are differentiable nowhere. An explicit example may be given by a function of the form
$$ \sum_{k=1}^{\infty} a^k \cos (b^k \pi x) $$
Where we take $ 0<a<1 $ (in order to make the series converge) and $ ab $ sufficiently large; for further references, look for Weierstrass's function on the web. He was the first to discover this kind of phenomena.
A reference for you if you wish to study more about this stuff/justify why the function I mentioned is nowhere differentiable may be this nice survey in form of a master's thesis: https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf
