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January 1st, 2012, 06:15 PM   #1
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uniform convergence of continuous functions.

Hi, here's my question:

Let be continuous, and . Let . I claim then that does not converge uniformly on and that .

Attempt at proof:
So the first bit isn't bad; either it can be shown by contradiction (suppose converges uniformly, and get a contradiction), or by noting that if did converge uniformly, then , which is a discontinuous function (since ). This contradicts the fact that if a sequence of continuous functions converges uniformly, then their limit is continuous as well.

As for the second bit, I'm having some trouble with the details. I want to do something like, for some :

but not quite sure how to proceed.

Thanks for any help!
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January 2nd, 2012, 12:35 PM   #2
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Re: uniform convergence of continuous functions.

The first integral (in your end expression) = f(0)(1-?). The second integral is bounded by ?M, where M = max(|f(x)|) over the interval. Continuity implies M is finite. Let ? -> 0 and get your result.
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