Does this figure represent a cumulative distribution function?

Question

Is this a c.d.f.?

I have no problem for random variable $X$ at $-\infty<X<x_2$. But if p.d.f. were continuous in interval $x_2\leq X<\infty$ , then c.d.f. should have been continuous. If probability mass function were discrete at $x_3$, then the c.d.f. would have a jump at $x_3$, but would remain constant thereafter upto $x_4$. The given figure does not have these both properties. Is this figure an example of c.d.f.? If yes, how is it?

(Definition: The c.d.f. $F$ of a random variable $X$ is a function defined for each real number $x$ as follows:$$F(x)=Pr(X\leq x) \text{ for } -\infty<x<\infty\text{ )}$$

Answer 1

This is a cumulative distribution function, because:

1. $F(x)\leq F(y)$ if $x \leq y$
2. $\lim\limits_{x \to {-\infty}}F(x)=0$ and $\lim\limits_{x \to {+\infty}}F(x)=1$
3. $F(x)$ is continuous to the right

You have a discrete variable at $x_3$, however the variable is continuous in $[x_3,+\infty)$ so it dones't have to be a constant.

That just means that $P\{x=x_3\}\ne0$

Answer 2

Does it satisfy the properties of a CDF? The properties being:

1. $F(x)\in [0,1]$ for all $x$.
2. $F$ is non-decreasing and right-continuous on $\mathbb{R}$.
3. $\lim\limits_{x\to -\infty}F(x)=0$ and $\lim\limits_{x\to\infty}F(x)=1$.

If so, then $F$ is indeed the CDF of some random variable $X$.

Remark: It does not need to be the CDF of a random variable that has density or is discrete (it can, for instance, be a mixture of the two).