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 a theory of "calculus on Banach spaces", which comes close to what you are asking for. See for example Wikipedia:Fréchet derivatives for some information about Fréchet and Gateaux derivatives for functions $f : U \to V$ where $U$ and $V$ are Banach spaces.
Some things remain the same in the infinite dimensional setting, but some things are fundamentally different. For example the closed unit ball in a Banach space is compact if and only if the space is finite dimensional.
