Suppose $f(x)$ is the probability density function of a random variable $X$, which means:

$$\int_{a}^{b} f(x) dx = 1$$

Also suppose $f$ is continuous and differentiable.

Provide a non-trivial condition under which $\int_{a}^{b} |f^\prime(x)| dx$ exists.

$[a,b]$ maybe a compact interval (regular integral) or $[-\infty, \infty]$ (improper integral).