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Uniform Continuity and Induction?
Let f: R-->R be defined by f(x)=x^k, for any natural number k. Determine what values of k, if any, make f uniformly continuous over R. If there does not exist such k, use induction to verify it.
1 Answer
- 9 years agoFavorite Answer
When k=1 the function is obviously uniformly continuous (delta = epsilon). Also true trivially for k=0. For k>=2 it is not uniformly continuous. In fact k=2 is the standard example of a simple function which is not uniformly continuous on all of R, and the situation just gets worse as k gets bigger, which means that you should first prove this for k=2 and then use induction for k>2.
