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Consider differentiable functions $g(s):\mathbb{R}\to\mathbb{R}_+$ and $h_\nu(x,s):\mathbb{R}^2\to\mathbb{R}$, for each $\nu\in\mathbb{N}$, and define

$$f(x,s)=\sum_{\nu=0}^{\lfloor g(s) \rfloor+1} h_\nu(x,s).$$

Obviously this function is not continuous everywhere but I am curious to know if, where it is continuous, it is also differentiable and what this derivative with respect to $s$ looks like. This is similar to the MSE post here but different in that the summand is also a function of $s$. I have tried setting up an analogous use of the Euler–Maclaurin Formula but I'm getting nowhere.

Any resources would be much appreciated.

Fisherman's Friend
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  • Outside $\mathbb R\times g^{-1}(\mathbb Z)$, $f$ is differentiable as a sum a fixed number of differentiable functions. If $g(s)$ is an integer, then $f$ is continuous on $(x,s)$ iff $h_{g(s)+1}(x,s)=0$. The differentiability might require more hypotheses (e.g. $C^1$). – Christophe Boilley Apr 07 '25 at 11:28

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