I do not believe that “any monotonically increasing bounded function over the reals is continuous”. For instance, choose some montonically increasing function bounded to (0,0.4) for x<-1, another function bounded to (0.45,0.55) for −1<x<1, and a third function bounded to (0.6,1) for x>1.
You’re right. I Misremembered, but i checked and it is true that a bounded montonic function of the reals can have only a countable number of discontinuities. So if ROB knows our algorithm, he can select one continuous interval for all of his values to come from.
Proof of the countable nature of discontinutes given here:
Thanks!
I do not believe that “any monotonically increasing bounded function over the reals is continuous”. For instance, choose some montonically increasing function bounded to (0,0.4) for x<-1, another function bounded to (0.45,0.55) for −1<x<1, and a third function bounded to (0.6,1) for x>1.
I did not check the rest of the argument, sorry
You’re right. I Misremembered, but i checked and it is true that a bounded montonic function of the reals can have only a countable number of discontinuities. So if ROB knows our algorithm, he can select one continuous interval for all of his values to come from.
Proof of the countable nature of discontinutes given here:
https://math.stackexchange.com/questions/2793202/set-of-discontinuity-of-monotone-function-is-countable