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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.

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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 $x=1$
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  • $\begingroup$ 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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