{$\Phi_s(x)$}$0 \leq s \leq t$ is a set of continuous and increasing functions which are uniformly bounded by $[0,1]$.Defind $\Psi(x) = \inf_{0 \leq s \leq t} \Phi_s(x)$,can we have $\Psi^{-1}(x) = \sup_{0 \leq s \leq t} \Phi_s^{-1}(x)$? IF this is right,how to prove it?
I have prove that $\Psi^{-1}(x) \geq \sup_{0 \leq s \leq t} \Phi_s^{-1}(x)$,but the provement of $\Psi^{-1}(x) \leq \sup_{0 \leq s \leq t} \Phi_s^{-1}(x)$ is hard for me.IF the continuity is uniform,it is easy to prove.Unfortunately,this is not allowed.
I'd appreciate it if someone could prove it. Any relevant hints are also welcome.