Assumptions for derivative of an integral

Question

for example if: $$F(x) = \int_{a}^{x} f(t)dt$$ and I want to use the fact that: $$F'(x) = f(x)$$ Do I simply need to show that f(t) satisfies the assumptions that it is continuous and bounded above and below, are there any other assumptions I missed?

Answer 1

If $I$ is an intervall in $ \mathbb R$, if $f:I \to \mathbb R$ is continuous and if $a \in I$, then

$$F(x) := \int_{a}^{x} f(t)dt \quad (x \in I)$$

is differentiable and $F'=f$ on $I$. That $f$ is bounded on $I$ is not needed.

Example: $I=(0,1]$ and $f(x)=1/x.$

Answer 2

It is sufficient to show that $f(t)$ is continuous (then, on any closed bounded interval, it will automatically be bounded above and below). See http://www2.clarku.edu/~djoyce/ma121/FTCproof.pdf.