Mathematicians don't argue about what an expression "really is" (or at least, real mathematics doesn't involve this). They define functions and use axioms to prove theories about them.

"No really". Mathematics just isn't concerned with this stuff. Sometimes infinity it defined as single point making the real number compact, sometimes a "positive infinity" and a "negative infinity" are defined. Sometimes you add points to a given function to make it more tractable and sometimes you don't. But none of this "means" anything. The real number line can be embedded in a number of topological spaces. At least two division rings and various things (the complex numbers are most common). The way you extend a given function (say e^x) is going to vary depending on what space you're looking at as well as what topic you're interested in.

Math works with definition systems and get theorems out of them. If you want to know what something "really is", consult philosophy or something.