# Absolutely Continuous Random Variable with Discontinuous Density

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: $$\lim_{x\rightarrow a^+}f(x)\neq\lim_{x\rightarrow a^-}f(x)$$.

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.