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Is there any real-valued function $f(x)$ of a real variable $x$ with $n^{th}$ and $m^{th}$ derivatives never equal for nonequal nonnegative $m$ and $n$ and where the $n^{th}$ derivative of $f$ never has a vertical asymptote?

The idea is that f should be infinitely differentiable like $e^x$ or $sin(x)$ without being periodic by differentiation. In this case I would call $e^x$ 1-periodic by differentiation and $sin(x)$ 4-periodic by differentiation.

The asymptote requirement is because the solution $f(x)=\frac{1}{x}$ is obvious.

Also, is there a name for this class of functions?

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3 Answers 3

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$f(x)=\frac{1}{1+x^2}$ should do the trick.

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  • $\begingroup$ Yes, it does. Now I am wondering if there is a name for these kinds of functions. $\endgroup$
    – mikebolt
    Feb 25, 2014 at 3:27
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    $\begingroup$ Having periodic derivatives is actually the much rarer case; it's just a rarer case that a lot of the standard elementary functions happen to fall into. $\endgroup$
    – Micah
    Feb 25, 2014 at 4:03
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    $\begingroup$ Having derivatives is a rare case that many functions we study fall into. $\endgroup$ Feb 25, 2014 at 4:22
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I like $xe^x{}{}{}{}{}{}{}{}$.

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$e^{2x}$ is not so bad either.

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  • $\begingroup$ Right, but that's no so interesting. You just end up with a scalar multiple of the function. $\endgroup$
    – mikebolt
    Feb 25, 2014 at 3:47
  • $\begingroup$ I guess you like $e^{x^2}$ better. $\endgroup$
    – LeoTheKub
    Feb 25, 2014 at 3:50

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