For a given function with certain properties we have that

$$ f^{(k)}(x) = \left(\frac{\mathrm d^kf}{\mathrm dx^k}\right) (x) $$

However for fixed $f$ and $x$ one could be interested in studying the function

$$ g_{f,x}(k) = f^{(k)}(x) $$

Which I guess it is well defined when $k$ is an integer (positive provides derivatives and negative provides integrals). I was wondering if there's some studies about such functions but instead of $k$ we have a real number (i.e. generalization of integral order and derivative order). Something like

$$ g_{f,x}(y) = f^{(y)}(y) $$

where $y \in \mathbb{R}$. It's more a curiosity rather than anything else.


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