Assume you have a random varaible with a cumulation distribution function $F(x)=P(X\le x)$. This function will always be right-continuous, but assume that it also is left-continuous. Then there are no single values which have a positive probability.
Will this random variable always have a probability density function which can be integrated?
By the Radon-Nikodym theorem what we have to show is that the measure induced by $F(x)$ is absolutely continuous with respect to the Lebesgue-measure. However, I do not know how to show this.