Finite convex functions are known to enjoy remarquable properties, but I found the word ‘’finite’’ a little bit ambiguous. I would like to know which is correct :
A convex function $f$ is finite if $f$ is finite-valued, i.e., $-\infty\lt f(x)\lt+\infty$ for all $x$.
Or
A convex function $f$ is finite if it takes finitely many values $y$ such that $f(x)=y$.
A finite convex function is a convex function whose output is never equal to $+\infty$ or $-\infty$. For example, $f(x) = x^2$ is a finite convex function.
I think restricting the set of possible output values of the function to be a finite set would rule out most interesting examples of convex functions.