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I have this function $f(x)$ which is continuous and differentiable on $\mathbb R$. Is the following true - without the assumption that $f$ being absolutely continuous!

\int_{a}^{b} f'(x) dx=f(b)-f(a)

Edit: $f$ is infinitely differentiable on $\mathbb R$. I think this may change everything!

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No, the derivative need not even be integrable! See Wikipedia's article on Volterra's function.

PS: If $f$ is differentiable, then $f$ is continuous.

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    Wait: Volterra's function is not *Riemann* integrable. Since the derivative is bounded and Borel, it evidently is Lebesgue integrable. By mentioning absolute continuity, I think this question is about the Lebesgue differentiation theorem which states that the fundamental theorem of calculus holds *precisely* for the absolutely continuous functions.2011-10-20
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After edit, it became true, because of continuously f'(x). And now you can use Newton-Leibniz formula

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    @Ashley It is. The formula holds iff $f$ is absolutely continuous.2011-10-20