Thursday, 25 April 2019

real analysis - Let f(x) be Riemann-integrable, F(x)=intxaf(t)dt. Then F(x) is differentiable, and F(x)=f(x) almost everywhere.


What is wrong with the following statement:



Let f(x) be Riemann-integrable, F(x)=xaf(t)dt. Then F(x) is differentiable, and F(x)=f(x) almost everywhere.




I think the statement is true. Because f(x) be Riemann-integrable, it's continuous almost everywhere. Then by the Second Fundamental Theorem of Calculus, F(x) is is differentiable almost everywhere, and F(x)=f(x) almost everywhere.

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