Let be continuous on . Prove that exists is uniformly continuous on .

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- Aug 11th 2009, 05:14 PMKat-Muniform continuity
Let be continuous on . Prove that exists is uniformly continuous on .

- Aug 11th 2009, 07:48 PMsiclar
If we know exists and we want to show f is actually uniformly continuous, it helps to invoke a nice theorem.

**Theorem**: If is a continuous real function on a closed and bounded set of real numbers, then is in fact uniformly continuous.

Then if we can define a value of at which maintains continuity, we've got it!

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Conversely if the function is uniformly continuous then a straightforward use of the definition shows the limit exists.

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