Improper integral/Bounded interval/Continuous extension/Existence/Exercise

Let ${}I={]r,s[}$ be a bounded open interval and

$f\colon I\longrightarrow \mathbb {R}$

a continuous function, which can be extended continuously to ${}[r,s]$. Prove that the improper integral

$\int _{r}^{s}f(t)dt$

exists.

Create a solution