Continuity of cumulative probability function

Question

Define $\hat F(t) = \inf\{x: F(x) > t\}$. Assume $F$ is a distribution function. Is $\hat F$ right-continuous on $(0,1)$?

Answer 1

Let $G_t = \{ x | F(x) >t \}$, note that if $s<t$ then $G_t \subset G_s$. Hence $\hat{F}$ is non decreasing.

Suppose $t_n \downarrow t$ and let $\epsilon >0$. By definition of $\hat{F}(t)$, there is some $x$ such that $x< \hat{F}(t) + \epsilon$ and $F(x) > t$. Now choose $N$ such that $F(x) > t_n$ for all $n \ge N$. Since $x \in G_{t_n}$, we see that $\hat{F}(t) \le \hat{F}(t_n) \le x< \hat{F}(t) + \epsilon$ and hence we see that $\hat{F}(t_n) \downarrow \hat{F}(t)$.