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.?  

There is no equivalent to Lebesgue Measure, which is used to define volume and hence integration on $\mathbb R ^n$ for $n \lt \infty$, for infinite dimensional spaces. Formally this means that there is no measure on any infinite-dimensional Banach $B$ space which is:
1) Locally finite: Each point hasan open neighborhood with finite measure.
2) Strictly positive: Every nontrivial open subset has strictly positive measure.
3) Translation Invariance: If $E \subseteq B$ and $\mu(E) = L$ then $\mu(E+x) = L  \ \forall x \in B$.
This is proved by showing that we can exploit the infinite-dimensionality of the unit ball to pack infinitely many balls of radius $\frac{1}{4}$ inside it. Which leads to the conclusion that the unit ball has infinite measure.
So there is no directly analogous way to define definite integration for functions with infinite-dimensional domain.
