Will the rules of calculus stay the same when a real-valued function is defined over infinite number of variables? So the question would be:


*

*Can we ever talk about a real-valued function that is defined over infinite number of variables?

*Will the rules of calculus remain the same for such functions described in 1.?  

Regarding 1, of course we can. Consider $X:= \mathbb R^{\mathbb N} = \{f: \mathbb R\to \mathbb N\}$, the set of all sequences of real numbers. It's perfectly reasonable to want to talk about functions from $X$ to $\mathbb R$. Or if you don't want to have to deal with the fact that a function $f: X \to \mathbb R$ depends on "infinitely many variables", you can specialise to $c_c = \{(x_n)_{n\geq1} \in X \mid x_n$ is eventually $0\}$.
Unfortunately $X$ (and $c_c$ for that matter) is not all that well-behaved. While it does have a natural topology which is even metrisable, it's missing many of the nice properties that makes the set of real numbers so useful—for example it's not locally compact and it admits linear maps into the reals that are not continuous.
