I have seen a nice number of theorems that start with "suppose that $f$ is continuous function" or with some equivalent claim and then, with only that, or with some additional assumptions some theorem follows.
But I would like to know about theorems that start with the assumptions like "suppose that $f$ is discontinuous function" and then end with some truth about discontinuous functions.
Suppose that we work with real functions of a real variable.
Because the function can be discontinuous in a finite number of points, in countably infinite number of points and in uncountably infinite number of points let us talk here only about functions that have uncountably infinite number of discontinuities.
So the question is:
Can you give me some examples of theorems that start with the assumption that "$f$ is real function of a real variable which has an uncountable number of discontinuities" (and possibly with some other assumptions) and then some fact about such functions follows?