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Uniform continuity example.

Is f: (0,1) -> R defined by f(x) = (x^2)cos(1/x) uniformly continuous?

I don't know any proof methods for uniform continuity and wondered if someone could explain it to me.

Thanks.
(edited 12 years ago)
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ttoby
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Original post by Ben77

Original post by Ben77
Is f: (0,1) -> R defined by f(x) = (x^2)cos(1/x) uniformly continuous?

I don't know any proof methods for uniform continuity and wondered if someone could explain it to me.

Thanks.


Use the mean value theorem http://en.wikipedia.org/wiki/Mean_value_theorem By differentiating, you can get upper/lower bounds on f'(c), and hence get an upper bound on |(f(b)-f(a))/(b-a)| which will work for any a and b.

You can then use this for an epsilon-delta proof, where you show that if |b-a| is sufficiently small then |f(b)-f(a)| is less than epsilon.

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