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Give an example of a uniformly continuous function f: (0,1) ---> R'

such that f' exists on (0,1) and is unbounded.

we came up with the example that f(x) = x*sin(1/x) if you interpret the question to mean f' is unbounded, not f itself.

Our teacher then asked us to ponder whether it is possible for the condition to be reversed, ie. what if the unbounded is refering to f(x) not f'(x). Is it possible for f(x) to be uniformly continuous but unbounded on the open interval.

The only thing I have thought would be if it had an infinite discontinuity at the end point. Like if the function went to positive infinity at x=1. Is it possible for what ever curve would do that to still satisfy the necessary conditions to be uniformly continuous?