Let f be a function that has an anti-derivative function F (F'(x)=f(x)) in [a,b].

Prove / Disproof :

f is integrable in [a,b].

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Well, I said it shouldn't be, and I gave the example:

$\displaystyle

F(x):= 0 , x=0$

$\displaystyle x\sin\frac{1}{x} , 0<x\le1$

$\displaystyle x\in[0,1]$

therefore,

$\displaystyle f:=\sin\frac{1}{x}-\frac{1}{x}cos\frac{1}{x} , x\in[0,1]$

f can't be integrable there, because it is not bounded for $\displaystyle x\in(0,1]$.

Is this a proper disproof?

Than you