A real differentiable function on $\mathbb{R}^+$ satisfying $f(x) = x!$ for $x\in\mathbb{N}$ and having minimal derivative $\left|\frac{\partial f}{\partial x}\right|$ everywhere, say in the sense that it minimizes $$\left\|\frac{\partial f}{\partial x}\right\|_k = \sqrt[k]{\int_{0}^\infty \left|\frac{\partial f}{\partial x}\right|^kdx}$$(For which k) will it be the (translated) gamma function?
(This is just a guess from me that there should be such a $k$.)
Edit as pointed out in the comments the integral above of course won't converge. Maybe we can replace it with $\left\|g\left(\frac{\partial f}{\partial x}\right)\right\|_k$ for $g(x) = \frac{1}{x}$ or $\log(x)$ or maybe switch to $\left\|\frac{dg(f)}{dx}\right\|$ like logarithmic derivative ($g=\log$) some other function guaranteed to be integrable. Or to limit the original measure to an interval or a sequence of intervals
$$\left\|\frac{\partial f}{\partial x}\right\|_k = \sqrt[k]{\int_{0}^{x_1} \left|\frac{\partial f}{\partial x}\right|^kdx}$$