# Simple question about the Fundamental Theorem of Calculus

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 $a\in\mathbb R$?

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

## 1 Answer

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: $f(x) = x^2 + 1$.