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.


Elementary functions, as usually defined, are continuos in all the domain of definition but not always differentiable. Let consider as an example of elementary functions continuous but not differentiable at a point

  • $f(x)=\sqrt x$ at $x=0$
  • $f(x)=\sqrt[3] x$ at $x=0$
  • $f(x)=\arcsin x$ at $x=1$
  • $\begingroup$ Thanks, I should have looked at these examples first, I feel dumb now $\endgroup$ – user63858 Jul 26 '18 at 16:47
  • $\begingroup$ It’s ok, now your overview is more clear I hope. You are welcome! Bye $\endgroup$ – gimusi Jul 26 '18 at 16:49

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