Anyone who knows the measure theoretic framework for random variables will know that there exists random variables that are neither discrete (i.e. taking countably many values) or absolutely continuous (i.e. having a probability density function). However, how does one go about proving that a random variable is not absolutely continuous?
Proving the negative that "this random variable does not have a probability density function" strikes me as a rather difficult task. I can imagine that there's some sort of proof by contradiction that can be done here, but I fail to see what contradiction can be drawn. After all, the definition of a probability density function is not very restrictive. Is there some equivalent property that I've missed?