Proving that the reciprocal of a Riemann integrable function is also Riemann integrable when its absolute value is greater than a positive number

Question

$ f$ is Riemann integrable on $[a, b]$ and $|f (x)| ≥ c > 0$ for $x$ ∈ $[a, b]$. Prove that $1/f$ is Riemann integrable on $ [a, b] $.

Any hints please ?

Answer 1

Use the fact that $f$ is Riemann-integrable if and only if $f$ is bounded and the set of those points at which $f$ is discontinuous has Lebesgue measure $0$.

Answer 2

For any nonzero numbers $u$, $v$ one has $$\left|{1\over u}-{1\over v}\right|={|u-v|\over |u|\>|v|}\ .$$ If $|f(x)|\geq c$ on $[a,b]$ we may therefore conclude that $$\left|{1\over f(x)}-{1\over f(y)}\right|\leq {|f(x)-f(y)|\over c^2}\qquad\forall\> x,\>y\in[a,b]\ .$$ Now take your favorite test for the Riemann integrability of ${1\over f}$ and connect it to the assumed Riemann integrability of $f$ via a $c^2$-blowup-factor.