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We learned and proved the Fundamental Theorem of Calculus, and spoke about functions defined using the theorem. I still feel I haven't quite understood the relation:

Let's say $f(x)$ is differentiable on $\mathbb R$ and $f'(x)$ is continuous on $\mathbb R$ .

Then can we say:

$f(x)=\int _{ a }^{ x }{ f'(t)dt }\ \ $ for some $a\in\mathbb R$?

If so, is there any significance to $a\in\mathbb R$?

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  • $\begingroup$ This takes $f(a)=0$. Then it is true. $\endgroup$ – Julien May 9 '13 at 16:53
  • $\begingroup$ @julien since $\int _{ a }^{ x }{ f'(t)dt }=f(x)-f(a)$? $\endgroup$ – Paz May 9 '13 at 16:55
  • $\begingroup$ Yes, precisely. When $f'$ is continuous, you get that by the FTC. $\endgroup$ – Julien May 9 '13 at 16:56
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As mentioned in the comments, we should pick a value of $a$ such that $f(a) = 0$. Then the statement is true.

This is important because, for some functions, picking a value of $a$ such that $f(a) = 0$ is impossible. For example: $f(x) = x^2 + 1$.

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