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September 11th, 2014, 08:16 AM   #1
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Fundamental Theorem of Calculus

Need help to solve the attached problem
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 September 11th, 2014, 08:34 AM #2 Member   Joined: Jan 2012 Posts: 51 Thanks: 1 I think I have a solution for letter a. Since f is differentiable at c , f is continuous at c and by FTC1 , F is diferenciable at c. Is it right ? For letter b, I tought this way: f is differentiable => f is continuous =>(by ftc) F is differentiable => F is continuous... (I can't see how I can continue from here, maybe its the case of a counter-example)
 September 11th, 2014, 10:00 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,659 Thanks: 2635 Math Focus: Mainly analysis and algebra I agree with you answer for a). For b) $F' = f$ wherever $f$ is continuous. Thus $f$ differentiable $\implies f$ continous $\implies F' = f$ For c) the existence of $f'$ mean that $f$ is continuous, so $F'=f$ and is therefore continuous and differentiable.

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