Suppose $f: [a,b] \rightarrow \mathbb{R}$ is Riemann intergrable on $[a+\epsilon, b]$ for all $0<\epsilon<b-a$.

Question

Suppose $f: [a,b] \rightarrow \mathbb{R}$ is Riemann intergrable on $[a+\epsilon, b]$ for all $0<\epsilon<b-a$. Then $f$ is Riemann integrable on $[a,b]$?

Answer 1

HINT:

$f(x) = \frac{1}{x}$ on $(0, 1]$ and $f(0) = 0$. The problem is : $f$ is not bounded.