Let be a compact metric space (feel free to impose more conditions as long as they're also satisfied by spheres) and a continuous function such that
1. for all
2. for all
Then, for each , let be the smallest such that . Is a continuous function?
A friend suggested that I applied the Maximum theorem, but to show that the relevant correspondence is lower semicontinuous I need to prove the following statement:
If and , there is a subsequence and a sequence such that and . This doesn't seem very obvious or even true, but I'm not sure.