My Math Forum Differentiability of Riemann Integrable Functions

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 April 15th, 2012, 11:01 AM #1 Senior Member   Joined: Sep 2008 Posts: 105 Thanks: 0 Differentiability of Riemann Integrable Functions Suppose f is Riemann integrable on [a,b] and let $F(x)=\int^x_a f(t)\,dt$ Is it possible that F?(x) exists but is not equal to f(x)? Is it possible that F?(x) fails to exist?
 April 15th, 2012, 02:12 PM #2 Global Moderator   Joined: May 2007 Posts: 6,235 Thanks: 497 Re: Differentiability of Riemann Integrable Functions F'(x) exists. The only caveat is what happens at points of discontinuity of f(x). There F'(x) is undefined. What you are asking about is the fundamental theorem of calculus.

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