
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
January 1st, 2012, 07:15 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, 01:35 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,642 Thanks: 625  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.


Tags 
continuous, convergence, functions, uniform 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
statistics uniform continuous distribution  najaa  Probability and Statistics  1  February 23rd, 2013 10:39 AM 
Show that a continuous derivative is a uniform limit  Jeh  Real Analysis  1  August 3rd, 2012 05:27 PM 
Uniform convergence  Series of functions  estro  Real Analysis  0  June 16th, 2010 05:35 PM 
Uniform convergence 2  rose3  Real Analysis  4  May 31st, 2010 09:11 AM 
Uniform Convergence  problem  Real Analysis  1  December 1st, 2009 08:00 AM 