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August 3rd, 2012, 07:46 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, 05:27 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 

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continuous, derivative, limit, show, uniform 
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