Is this definition of a quantile proper?

Question

I need to find a proper definition of a quantile. It says: a p-th quantile $x_p$ is a number, that satisfies the following conditions: $$ 0<p<1 $$ and $$ P(X \le x_{p}) \ge p $$ and $$ P(X \ge x_{p}) \ge 1-p $$ is this definition right?

Answer 1

Yes, except for $0<p<1$, I would use

$$0 \leq p \leq 1$$

because probability can bei $0$ and $1$, too. The other conditions are fine.

Answer 2

$$\text{Yes.}{}{}{}{}{}{}{}{}$$