I have some issues with the cumulative distribution function ( denoted by $F$ of a random variable $X$, his quantile $Q_X(\alpha)$ and the inverse of the cumulative distribution function $F^{-1}$.
Here my thoughts :
1) The inverse $F^{-1}$ does not always exists expecially if F is not bijective ( continuous and strictly monotonic and therefore strictly increasing because $F$ is a a cumulative distribution function) .
2) the quantile $Q_X(\alpha)=inf \{x\in \mathbb{R} \text{ such that } F(x)\le \alpha \} $ always exits and is well define.
3) if $F$ the cumulative distribution is strictly increasing therefore $Q_X=F^{-1}$.
Are my thoughts true or not?
Thanks in advance.