# Are Cumulative Distribution Function for a Continuous Random Variable Left Continuous also?

I was reading the book "probability random variables and stochastic processes by Athanasios Papoulis - Third Edition" where by in the properties of distribution functions on Page 69 of the book the fifth(5th) point mentions that Cumulative Distribution Function are right continuous in general(for discrete and continuous random variable). But I have a doubt whether we can say that:

Cumulative Distribution Function are both left and right continuous for continuous random variables in general or not?

• The CDF $F$ of a continuous distribution is not only continuous, but also absolutely continuous, in the sense that $$F(x) = \int_{-\infty}^{x} f(t) \, \mathrm{d}t$$ for some integrable function $f$. In this case, of course $f$ is the PDF of the distribution. For the record, note that there are distributions which are not continuous but have continuous CDFs. (The quintessential example is Cantor distribution.) – Sangchul Lee Sep 1 '19 at 4:14
• @SangchulLee : There is a question of convention: Should one define "continuous distribution" simply to mean one whose c.d.f. is continuous, or should one define it to mean a distribution for which there is a density? If the later, then you are right. I have come to think that the former is the better convention. Either way, there are some distributions for which the c.d.f. is continuous but the distribution neither has a density nor is a mixture (i.e. a weighted average) of any distributions with densities and any other distributions. – Michael Hardy Jan 1 at 14:53

The cumulative distribution function for a continuous random variable $$X$$ is continuous since $$X$$ admits a density $$f$$. Hence $$F$$ is of the form $$F(x)=\int_{-\infty}^xf(t)\,dt$$ which is continuous.