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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?

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The answer is yes, there is another type of random variable. The Lebesgue Decomposition Theorem fully categorizes all probability measures on the real line, and breaks them down into three types. Two of them are the types you mentioned. An example of the third type is the Cantor Distribution. Taken from the wiki, it is "the probability measure on the real line whose cumulative distribution function is the Cantor function".

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