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I am curious if anyone can construct a function made up of more than one e.g. $|x| = x, x\geq 0$ and $-x, x\leq 0$. However I would require that it must be infinitely differentiable and in the above case of $|x|$ it must be infinitely differentiable at 0. I suspect that we can't find one but if anyone has an example or a rigorous explanation as to why it doesn't work it would be grateful.

user61038
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  • What do you mean "made up of more than one"? Are you looking at piecewise defined functions? In such a case, you need each piece to be differentiable, and that side derivatives coincide at the gluing points. In your case, $x$ and $-x$ do not fill the bill. – Pedro May 08 '13 at 01:11
  • ofcourse, sorry I forgot the term for a brief second. Yes you do need something at the gluing points, but I want an example where they glue infinitely many times – user61038 May 08 '13 at 01:18
  • at the peaks of what specifically? are you saying at the joining points do some form of a convolution or what? – user61038 May 08 '13 at 02:41
  • (I meant the peaks of $\sin (x)^n$, n being an exponent) – Wolphram jonny May 08 '13 at 04:45

1 Answers1

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The standard example is $$f(x)=\left\{\begin{array}{ll}e^{-1/x},& x>0\\ 0,& x\le 0\end{array}\right.$$ $f$ is infinitely differentiable everywhere.

TCL
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