Def.

A random variable $X$ is said to be continuous if its cumulative distribution function has the form

$$F_X(x) = \int_{-\infty}^{x} f_X(\xi) d\xi$$

$f_X$ is called the probability density function (briefly density) of $X$.

Clearly

$$f_X(x) \ge 0, \quad \int_{-\infty}^{+\infty} f_X(\xi) d\xi = 1$$

for any density function $f_X$.