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Let $X$ be an absolutely continuous random variable. We denote by $f(x)$ its density (RN derivative). I know that $f(x)$ is almost everywhere defined, so $f(x)$ and $g(x)$ can both be the RN derivative with $g(x)\neq f(x)$ countably many times (it can jump countably many times).

My question is if the density of $X$ can have a discontinuity in the sense that: $\lim_{x\rightarrow a^+}f(x)\neq\lim_{x\rightarrow a^-}f(x)$.

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A density function is a non-negative measurable function $f:\mathbb R \to \mathbb R$ such that $\int f(x)dx=1$. Any such function is the density function of some random variable. An example of a discontinuous density function is $f=I_{(0,1)}$.

Note that if $f=g$ almost everywhere and $g$ is continuous then $f=g$ on a dense set so $g(0-)=0$ and $g(0+)=1$ leading to a contradiction.

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