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August 3rd, 2012, 06:46 AM   #1
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Show that a continuous derivative is a uniform limit

Suppose is differentiable on , where the derivative at an endpoint is the one-sided derivative, and suppose that the derivative is continuous. Show that the sequence of functions
converges uniformly to on any compact subinterval of .
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August 3rd, 2012, 04:27 PM   #2
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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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