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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?

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    $\begingroup$ 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.) $\endgroup$ – Sangchul Lee Sep 1 '19 at 4:14
  • $\begingroup$ @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. $\endgroup$ – Michael Hardy Jan 1 at 14:53
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There is a serious conflict in terminology as far as continuous random variables are concerned. Well known authors like Chung, Breiman, Feller etc do not agree that a continuous random variable has a density. It is mostly in non-rigorous or non-measure theoretic treatments that this terminology is used. In rigorous treatments of Probability Theory a random variable is said to be continuous if its distribution function is a continuous function and it is absolutely continuous if the distribution function is absolutely continuous which is true iff there is a density.

I summary, the answer to your question depends on what you mean by a continuous random variable.

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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.

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