In calculus when we look at $x=g(t), y=h(t)$ we first ask if this is a function $y=f(x)$. Then we look at properties like :
What is the domain of X and Y?
- Is it continuous, monotonic, smooth, analytic?
- Asymptotic behavior, special points like poles (e.i. $0$ for $y=1/x$)
- Cusps
- What is the domain of the Tailor series expansion?
- The norm and its implications for Fourier analysis
Let's assume I have a new probabilistic function that I want to study. It is nice in the sense of continuity and smoothness, but I don't have an analytic solution for it.
I can compute f(x) for a given x in seconds. What should I study to make it a well-defined function?
This function arises from some probability distribution. So, the integral over the domain of X is exactly 1. I use "well defined" in a broad sense: "I have a recipe to compute f(x) and the value stays the same". Here is my question. What should I study to characterize this function?