What is the concept of pathological function?

Question

mixed derivative theorem: Mixed partial derivatives Fxy and Fyx are always equal except for pathological functions.

for using mixed derivative theorem function must be non-pathological,so i want a way to find out a function is pathological or not?

Answer 1

I don't think there is any definition of pathological relating to functions (or any other mathematical concept). It's simply a word used informally to describe a function that behaves in a way you might not expect and thereby serves as a counterexample to some claim you might think was true. But a function one person finds pathological because it's "weird" (that's not defined either) in relation to his studies/interests might be uninteresting to another person, because it doesn't offer anything in relation to his studies/interests.

Answer 2

You can think pathological as counterintuitive. Like everywhere continuous but nowhere differentiable function, uncountalbe null set, banach-tarski paradox. And it's not the rigorous concept, so it's weird to say finding out some function is pathological or not.