Quantile function confusion about $\min\{x \text{ | }F(x) \geq p\}$

Question

I am currently reviewing the quantile function for discrete random variables but I am a bit confused.

We use the following definition of the quantile function: $$ \tilde{x}_p = \min\{x \in \mathbb{R}\mid F(x) \geq p\} $$ where $$ F(x)=\operatorname{Pr}(X \leq x) $$

Why is it required to have $F(x) \geq p$ if $F(x)$ only gives the probability of $\{X \leq x\}$?
The $\text{greater than}$ relation confuses me.

Because the Distribution function gives me the probability that my discrete random variable $X$ has a value less or equal to some value $x$.

So I would assume that the quantile function has to give a $x$ for a given percentage $p$ which is less or equal than $p$.
But the definition says that the $x$ has to be the smallest value so that $F(x)$ is greater than or equals $p$?

Does it have something to do with the random variable being $discrete$?
If yes could someone elaborate that?

I read different textbooks but even those have different definitions of the quantile.

Answer 1

Consider the random variable $X$, which we define to be constant $0$. It has the distribution $$ F(x) = \begin{cases} 0 & \text{if } x\leq 0 \\ 1 & \text{else} \end{cases}$$

Define the quantile function $$F^{-1}(\alpha) = \inf \{ x\in\mathbb{R} \,: F(x) \geq \alpha \}$$

Consider the median of $X$. It is supposed to be $F^{-1}(1/2)$. We know by intuition that it should be $0$, since that is the only value that $X$ will ever take.

Now compare these sets and their infimums \begin{align} S_0 &= \{x\,: F(x) = 1/2 \} = \emptyset &&\inf S_0 = -\infty \\ S_1 &= \{x\,: F(x)\geq 1/2\} = [0,\infty) && \inf S_1 = 0 \end{align}

So if you only allow equality, and not the greater-than-or-equal relation, you get the wrong result.

This fact stems from the discontinouities in the distribution function $F(x)$ and the convention that it is right-continous. If the distribution function was left-continous, we could do some other variant of the quantile function as well.