
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 3rd, 2012, 06:46 AM  #1 
Newbie Joined: Jul 2012 Posts: 2 Thanks: 0  Show that a continuous derivative is a uniform limit
Suppose is differentiable on , where the derivative at an endpoint is the onesided derivative, and suppose that the derivative is continuous. Show that the sequence of functions converges uniformly to on any compact subinterval of . 
August 3rd, 2012, 04:27 PM  #2 
Senior Member Joined: Jul 2011 Posts: 118 Thanks: 0  Re: Show that a continuous derivative is a uniform limit so the limit exists in Using mean value theorem: This is valid only for any interval as must lie in so that derivative exists As is continuous in closed interval it is uniformly continuous there. So for any we can find such that if then . Thus for we get for all 

Tags 
continuous, derivative, limit, show, uniform 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
D[f] is bounded, show that f is uniformly continuous  king.oslo  Real Analysis  1  July 7th, 2014 04:30 AM 
statistics uniform continuous distribution  najaa  Probability and Statistics  1  February 23rd, 2013 09:39 AM 
show (x+2)/(x^2x) is continuous (epsilondelta)  watson  Real Analysis  7  January 5th, 2012 07:43 PM 
uniform convergence of continuous functions.  watson  Real Analysis  1  January 2nd, 2012 12:35 PM 
show sequence of functions contains uniform convergnt subseq  watson  Real Analysis  6  December 26th, 2011 01:27 PM 