I wanted to ask something that I wasn't able to find online, and certainly haven't come across in my undergrad studies. Recently, I was solving a problem which had a function of the form $$ S(n) = \sum _{k=1}^{n}f(n-k). $$ where $f:\mathbb{R} \rightarrow \mathbb{R}$. Notice here that the variable $n$ appears both inside the function $f$ and as a limit in the summation. My question concerns the following:
Certainly, $S$ is a function from the natural numbers to $\mathbb{R}$. In this sense, $S$ is not continuous over $R$ in the usual notion $$ \lim _{n \rightarrow c}S(n) = S(c), $$ for $c \in \mathbb{R}$. But still, after plotting out the values of $S(n)$ for natural values, the figure would much resemble a curve. Given this, is it possible to assign a meaningful value to $\partial _n S$? (assuming instead that $S$ be replaced with a function whose values at natural numbers coincide with those of $S$).
I'm pretty sure someone must have come up with this before, but I cannot seem to find any papers on it. I'd gladly appreciate if someone could point me in the right direction, mention some notable texts, and give a brief overview of how such a thing is accomplished. Thanks! :)