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January 1st, 2012, 06:15 PM  #1 
Senior Member Joined: Nov 2011 Posts: 100 Thanks: 0  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! 
January 2nd, 2012, 12:35 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,583 Thanks: 610  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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continuous, convergence, functions, uniform 
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