I tried to solve it by cases: domain is a set of numbers; domain is an interval,;domain is a union of numbers and some intervals.
For the first case, I thought about arctanh is unbounded but its domain is bounded. To make it uniformly continuous, I can let Z be the domain.
For the second case, I think there does not exist a function like this.
For the third case, I am not sure if there exists a function satisfying all these conditions..
Did I think anything wrong for this question? or Could you give some idea or hint about that?