If $f:\mathbb{R}\to\mathbb{R}$ is continuous almost everywhere and $X$ is a random variable, then is $f(X)$ a random variable?

Question

I am currently working on a homework problem and I believe I need the result in the title. If $f$ were continuous, then the preimage of an open set, $f^{-1}(U),$ would be open and thus, $X^{-1}\left(f^{-1}(U)\right)$ would be measurable and thus, I could conclude that $f(X)$ is a random variable. However, I am not sure if continuous almost everywhere changes anything.

Answer 1

A counter-example: There exists a subset of the Cantor set which is not a Borel set. On $\mathbb R$ with the Borel $\sigma-$ field and, say a Gaussian probability measure, consider $I_E$. Let $X$ be the identity map on $\mathbb R$. Then $X$ is a r.v. but $I_E(X)$ is not, even though $I_E$ is continuous on $\mathbb R \setminus C$, hence continuous a.e.