September 11th, 2014, 08:16 AM  #1 
Member Joined: Jan 2012 Posts: 51 Thanks: 1  Fundamental Theorem of Calculus
Need help to solve the attached problem

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 counterexample) 
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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