Theorems about functions with uncountable number of discontinuities 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?

 A: If a function $f : (a,b) \to \Bbb R$ has an uncountable number of discontinuities, then only a countable number of them may be jump discontinuities, the others (uncountably many) being essential discontinuities.
If a function $f : (a,b) \to \Bbb R$ has an uncountable number of discontinuities, then it cannot be monotonic.
A: One result that comes to mind is the fact that at most countably many points can be points at which both the left limit and the right limit exist (finitely or infinitely) and are different from each other and are different from the value of the function at that point. Thus, if $f:{\mathbb R} \rightarrow {\mathbb R}$ has uncountably many discontinuities, then at all but countably many of these discontinuities we must have at least one of the unilateral limits not existing (finitely or infinitely).
In fact, there are much stronger statements that can be made -- see my answer to the mathoverflow question A search for theorems which appear to have very few, if any hypotheses.
A: No.  There is nothing that can be said about discontinuous fuctions.  Nothing at all.  Any pairing of real numbers to any other real numbers with no rationale, pattern, predictability, no determination or dependence can be a function, so there is nothing that can be concluded by knowing a function exists.
With one exception...
If f is a function that maps X to Y and $x \in X$ we can conclude there exists an $f (x) \in Y$ but there is noting we can say about what f (x) might be or any relation f (x) might have with any other y and f (y).
But that's the only thing we can say.
A: One such example is the Lebesgue's Criterion for Riemann Integration. 
In context of your question let $f$ be a bounded function defined on a closed and bounded interval such that its set of discontinuities is uncountable and of measure 0, then $f$ is Riemann Integrable. 
You can read more about the Lebesgue's Criterion here, and can find an example of such a function here.
