TITLE: Does the quantile function uniquely determine the distribution function?
QUESTION [7 upvotes]: For a probability distribution, its quantile function is defined in terms of its distribution function as 

$$ Q(p)=F^{-1}(p) = \inf \{ x\in R : p \le F(x) \} $$ 

I was wondering if, conversely, a quantile function can uniquely determine a distribution and therefore fully describe the probability distribution just as a distribution function does?
Thanks and regards!

UPDATE:
Please let me be more specific. Because a CDF is nondecreasing, right-continuous and limit is $0$ when $x \to -\infty$ and $1$ when $x \to \infty$, its quantile function is nondecreasing, left-continuous and a map from $(0,1)$ into $R$. If a function is nondecreasing, left-continuous and a map from $(0,1)$ into $R$, can it become a quantile function of some CDF? When it can, is there a way to represent the CDF in terms of the quantile function using infimum or supremum similar as quantile function in terms of CDF?

REPLY [2 votes]: I believe the following definition for a CDF is consistent with the definition of a quantile function in your original post:
$F(x) = \sup \{ p\in (0,1) : x \ge Q(p) \}$
This definition indeed makes the quantile function left-continuous as you proposed.