Integrable on subintervals but not integrable on the whole interval

Question

Let $f$ be a complex valued function on the compact interval $[a,b]$. If $f$ is integrable on $[a,s]$ for each $s$ in $(a,b)$, and $\lim_{s \to b^-} \int_a^sf$ exists, is $f$ necessarily integrable on the whole interval $[a,b]$? The answer is written to be no, but I cannot find an appropriate counterexample. Could aynone help me?