OK, I know that a random variable $X$ from some probability space to $\mathbb R$, with some additional properties. It is discrete if it's image in $\mathbb R$ is dicrete. It is otherwise called continuous, which can possibly mean that it's values can be the arbitrary intervals or the all real numbers. The formal definition of continous random variable $X$ says that there is a function $f : \mathbb R \rightarrow \mathbb R$, such that $f \geq 0$ and for every $a \leq b$
$$P(a \leq X \leq b) = \int_a^bf(t)dt$$
I understand that measuring the probability of $X$ to be a certain point is useless and that's why the interval $а \leq X \leq b$ is used. So my question is, if for every interval $I \subseteq \mathbb R$ we know the probability of $X$ to be inside $I$, is there always an apropriate function $f$ ? In other words is there another type of random variables for which such function does not exist?