I am studying for a qualifying exam and have come across this problem from a past exam. I would like to know how to complete it if anyone can help.

We have $\displaystyle f\colon [0,1] \to \mathbb{R}$ given as a Riemann integrable function whose values are all greater than 1, and must show that 1/fis also Riemann integrable.

I have shown the case when the originalfis bounded, as Riemann integrability implies continuity almost everywhere, which carries to 1/f(which is also bounded) and is hence Riemann integrable.

But I cannot reckon the situation involving an unboundedfwithout appealing to the ugly partition definition of Riemann integrability. Is there a nicer way? I am inclined to think so because it's a qual problem designed to be completely solved in twenty minutes.