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Lately I have been asked if there exist any two continuous random variables, whose sum is discrete random variable. Probably the good start is the definition of discrete and continuous random variables.

If random variable has property $\sum_{x \in R} P(X=x) = 1$, the variable is discrete.

Random variable is continuous if there exists function $f_X: \mathbb{R} \rightarrow \mathbb{R}_{\geq 0}$ which for each $A \subseteq \mathbb{R}$, $P(X \in A) = \int_A f_X(x) \ dx$.

Than means $P(X = x) = 0$ for each $x$ if variable is continuous. That gives me feeling that sum of two continuous random variables cannot be discrete, but I have no idea if I am right or wrong.

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It is possible for the sum of continuous random variables to be discrete: if $X$ is a continuous random variable, then so is $-X$, and $X+(-X)=0$, which is a discrete random variable.

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    Oh, right, it is pretty simple. It wasn't part of the question, but now I am thinking about their independence - they are **not** independent, right? If so, are there two independent continuous random variables, whose sum is discrete? – 2017-02-12
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    No: if $X$ and $Y$ are independent with pdfs $f$ and $g$, then the sum $X+Y$ has a pdf given by the convolution of $f$ and $g$. – 2017-02-12
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    Complement: If $X, Y$ are independent and one of it has pdf, then the sum $X+Y$ has a pdf also. – 2017-02-13
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    Dumb question, but wouldn't the sum for all x over the 0 distribution still be 0 (not 1)? – 2017-02-13
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    I'm not sure what you mean. $0$ refers to the constant random variable zero, i.e. the random variable which is equal to zero with probability $1$. – 2017-02-13