Examine the uniform continuity of f : R given by f(x) := on .
I know it's not uniformly continuous but how do I prove that?
It is uniformly continous:
By the mean value theorem we have where . Taking the supremum over all and noting that we have that for all and so . Now taking in the definition of unif. cont. the result follows.
Actually you can prove that for a continously diff. function it's equivalent:
1) is bounded
2) There exists such that