Let $F(v)=\inf\{x: f(x)\leq v\}$, where $f(x)$ is a nonincreasing, right continuous function.
I have already prove that $F(v)$ is nonincreasing also. However, how to prove this is right continuous?
Right coninuous means if $v_m\to v,\ v_m\geq v$, then $F(v_m)\to F(v)$.
This is what I tried:
Pick a sequence {$v_m$} decreasing to $v$. We must have $F(v_m)$ monotone increasing, bounded by $F(v)$ So $F(v_m)$ converge to some number, say $F(\hat v)$. Now we just need to prove $F(\hat v)\geq F(v)$. If this is not the case, i.e.$F(\hat v)
Any suggestions?