Could you help me to understand if the following statements are correct?
Let $X$ be a random variable with cdf $F$ (so $F$ is, by definition, non-decreasing, right continuous, $\lim_{x\rightarrow-\infty}F(x)=0$, $\lim_{x\rightarrow\infty}F(x)=1$)
1) $X$ is a continuous random variable if and only if $F$ is a continuous functions
2) $F$ is a continuous function strictly monotone (increasing) if and only if $F^{-1}$ exists
3) $X$ is a continuous random variable and $F$ strictly monotone function (increasing) if and only if $F^{-1}$ exists