# Absolute continuity of distributions --- why do we need this? [duplicate]

Why do we define continuous probability distribution as those with absolutely continuous CDF, instead of just continuous CDF?

• Thanks, the link you provided is very helpful! Mar 18 '20 at 4:37

You can find more detail on this issue in a related question here. Absolute continuity of the CDF is a stronger condition than continuity, and essentially just means that the distribution has a valid density function. For example, if the CDF $$F$$ of a real scalar random variable is absolutely continuous then there exists a real function $$f$$ (the density) such that:
$$F(x) = \int \limits_{-\infty}^x f(x) dx.$$
• I"m curious what you mean by a "uniformly continuous distribution." One example I am puzzling over is any Beta$(a,b)$ distribution where $a$ or $b$ is less than $1.$