Let $X$ be a random variable with probability distribution $F$.
To define the question. We have to define to notions.
First, we define the points of support of $X$ as \begin{align} {\rm supp}(X)= \{x: \text{for every open interval } I \ni x \text{ we have that } F(I)>0\} . \end{align}
Now define the quantile functions as for $p \in (0,1)$ \begin{align} Q(p)= \inf \{ x: p \le F(x) \} \end{align}
My question is what is the relationship between ${\rm supp}(X)$ and the range of $Q$? Let's denote the range of $Q$ by $\text{Range}(Q)$.
I think we have the following \begin{align} {\rm supp}(X)= \text{Range}(Q). \end{align}
This is easy to show for discrete distribution with finitely many points and absolutely continuous distributions. Can we show this in general?
I think the key is that $Q(F(X))=X$ almost surely, but I am not entirely sure how to use this to show the desired result formally.