# Continuity and differentiability of elementary functions

Given a single-variable elementary function (not piecewise), I was wondering if it is continuous and differentiable in all of its maximum domain? (Not considering the trivial examples involving absolute value function)

Also by piecewise I mean functions that can only be expressed in a piecewise manner. E.g. $\cos x$ is not considered piecewise, although it can be expressed in a piecewise manner.

I think the definition of elementary function that I’m going for is the one found on wikipedia: https://en.m.wikipedia.org/wiki/Elementary_function I think it is also required for the function’s domain to at least contain some interval, but I’m not sure.

Actually I know that if an elementary function is differentiable at some point, its derivative at that point is also given by an elementary function, but I am trying to find out if every elementary function is smooth given that it is defined at some point.

I haven’t been able to find anything useful online, so a rigorous proof would be greatly appreciated. Also please try to keep it as simple as possible as my math is bad.

• $f(x)=\sqrt x$ at $x=0$
• $f(x)=\sqrt[3] x$ at $x=0$
• $f(x)=\arcsin x$ at $x=1$