0

If $f(x)$ is a non-atomic probability distribution function, then its integral $F(x) := \int_{-\infty}^{x}f(y)dy$ is a continuous function. However, if $f(x)$ has atoms then $F(x)$ may be discontinuous ($F$ might have jumps where $f$ has atoms).

What is an adjective weaker than "continuous" that correctly describes all functions $F$ that can be cumulative distribution functions, even when the probability distribution has atoms? Is it true that all such functions are "continuous almost everywhere"?

[Note: I am looking for a term specifically related to continuity; I ignore the fact that $F(-\infty)$ should be 0 and $F(\infty)$ should be 1].

Erel Segal-Halevi
  • 10,555

1 Answers1

1

A function $F:\Bbb R\to\Bbb R$ is the CDF of some probability - i.e. there is some probability $P$ on $\Bbb R$ such that $F(x)=P(-\infty, x]$ - if and only if:

  1. $\lim\limits_{x\to-\infty}F(x)=0$ and $\lim\limits_{x\to\infty}F(x)=1$

  2. $F$ is weakly increasing

  3. $F$ is right-continuous (i.e. $\lim_{x\to t^+} F(x)=F(t)$ for all $t$).

It is true that monotone functions $\Bbb R\to\Bbb R$, and thus those satisfying (2), have countably many discontinuity points. This is because of the fact that the family $$\left\{\left(\liminf_{x\to t} F(x),\limsup_{x\to t}F(x)\right)\,:\, t\in\Bbb R\land\ F\text{ discontinuous at }t\right\}$$ is a family if disjoint open intervals.