Prove or disprove: If $f$ is continuous on $(a,b)$, then $f(a)$ and $f(b)$ can be defined so that $f$ is integrable on $[a,b]$.

Question

I was wondering if $\tan x$ on the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$ is a valid response against the proposition. Irregardless of how we define the end points, we will still have an unbounded function and for the upper integral and lower integral to exist, we require a bounded function. Is this correct?

Answer 1

Yes, that is correct. For Riemann integrability boundedness is necessary. For Lebesgue integrability, your example shows that the statement is not correct as it stands.

Answer 2

Yes, your example is O.K. Another example: $f(x)=1/x$ for $x \in (0,1).$