Comment by skhunted

The word "function" in the analysis namespace defines it to ONLY EVER be a mapping S -> R or S -> C, where S is a subset of C^n or R^n. The word "function" is not allowed to be used - within this namespace - to denote anything else.

In real analyis one is interested in functions from R^n to R. They don't define function to be only something from R^n to R. It's just that these are the functions they wish to study. They don't define function to exclusively be a map from R^n to R. It’s just that these are the types of functions they care about.

No mathematician can possibly think function is anything other than a subset of the product of two spaces that meets a certain condition.