>>11773704> Real anal just seems like a way to justify calculusThis is not true. I will give you an example. In complex analysis you study holomorphic functions, in particular you get to know that a real part u(z)=u(x,y) of holomorphic function f(z) is a harmonic function. On the other hand you know that each harmonic function u(x,y) in some domain Omega has a unique (up to additive constant) harmonic conjugate, that is a function v(x,y) such that u+iv is holomorphic in Omega.
Just to recall, harmonic functions describe things like steady temperatures in the body, electric potential, divergent less, steady-state type flow. If u describes say the temperature in a body and v is its harmonic conjugate, then the curves v(x,y)=C describe the line of flow of the heat in that body. In other words, useful stuff.
So now you ask yourself what does it have to do with real analysis? Well consider the following problem. You have an unknown temperature distribution u on the upper half plane, but you know its values on the real axis. Question: can you find the temperature distribution on the upper halfplane? A classical result says that you can if f is continuous. Moreover since u has a harmonic conjugate v, you can define the function g(x)=\lim y\to 0 v(x,y). Now you can ask a series of questions: How do the properties of g depend on f? Does the same process work for f that is not continuous? If f_n is a sequance of functions of some class that is Cauchy in some sense is the same true for functions g_n?
To answer those questions you need real analysis, especially that cringy pedantism.
