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The answer to this question is most likely no, but I'm asking anyway: Assume that $f\in C^n(\mathbb {R,R})$. Is their any natural generalisation of the map $$\{1,2,\ldots,n\}\to C(\mathbb{R, R})\\k\mapsto f^{(k)}$$ to a map $$ [1, n]\to\{g\colon\mathbb{R\to R}\}\\ x\mapsto f^{(x)}? $$ That is, can we generalise "the $k$th derivative" to "the $x$th derivative" for real values of $x$? What I mean by "natural" is: Anything that has desirable properties and that has been explicitly formulated by someone, somewhere.

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    $\begingroup$ Of course. You can try looking for the keyword "pseudodifferential calculus". $\endgroup$ Commented Mar 2, 2014 at 10:44
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    $\begingroup$ There exists a whole industry of "fractional calculus" with different definitions of what a "fractional derivative" is. For instance, as mentioned in the previous comment, using the Fourier transform there is a natural extension from integer to fractional derivatives. $\endgroup$ Commented Mar 2, 2014 at 10:47
  • $\begingroup$ @Gaussler try seeing Calculus of Finite differences $\endgroup$
    – happymath
    Commented Mar 2, 2014 at 10:48
  • $\begingroup$ The Wikipedia article for fractional calculus mentions complex order derivatives too. No clue how that works. $\endgroup$
    – nog642
    Commented Jan 5, 2018 at 19:10

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