For every possible mapping on a set of real numbers to a set of real numbers, is there any overarching theory of their classification?
For example, some mappings aren’t defined by a formula but are just arbitrary mappings.
I’m not sure but I think every real function has to be defined in terms of operators that are themselves defined in terms of addition and multiplication, since those are the only two operations assigned to the field of real numbers. In other words, trigonometric functions and exponential functions have reformulations in addition and multiplication, if you can also use infinite limits.
Is there a known way of carving up the space of real functions by fundamental distinguishing properties or is it a partial classification in which certain subgroups are known, but they may partially overlap in irregular ways, and there may be currently unknown classes of functions?
As for why I want to know this, because I have been thinking about neural networks more abstractly and trying to consider what the abstract necessary characteristics are of an optimization function, to consider what valid alternatives to backpropagation could be by considering the space of all valid optimization functions; and to consider what abstract characteristics neural networks themselves have that allows them to do what they do; and similar questions.