TITLE: integral from zero to zero
QUESTION [7 upvotes]: it seems obvious that this integral is zero and so is the limit but what theorem we are using here?
I see it's connected to Riemann sums with an interval=zero Right ?
The function $\mathrm{f}$ is continuous.
$$\lim_{x \to 0}\int_0^x\mathrm{f}(x)\ \mathrm{d}x= \ ?$$

REPLY [2 votes]: Another approach valid when $f$ is continuos: the integral is a function $F$ s.t. $F′=f$ and $F(0)=0$. By continuity, $\lim F=F(0)=0$.