Is it possible that bounded function on a compact interval of $\mathbb{R}$ has infinite Riemann integral?

Question

Is it possible that a bounded function on a compact interval of $\mathbb{R}$ has infinite Riemann integral?

Answer 1

If the absolute value of the function is less than $M$ and the length of the interval is less than $l$, then none of the Riemann sums will exceed $Ml$ in absolute value. So, in short, no. This isn't possible.

On the other hand, there are plenty of bounded functions on compact intervals for which the limit which defines a Riemann integral won't exist. So a bounded function might fail to be integrable, but you won't get any infinities showing up.

Answer 2

No. If $f$ is bounded (and integrable) on $[a,b]$, then $$ \left|\int_a^b f \, dx\right| \leq \int_a^b |f| \, dx \leq (b-a)\max_{[a,b]}|f|. $$