Bounded variation of $\frac1f$ when $\inf(|f|)>0$ & $f$ bounded variation

Question

I want to show if $\frac{1}{f}\in BV[a,b]$

when $\inf(|f|)>0 \land f\in BV[a,b]$.

I tried to find a partition that $V(\frac{1}{f},P)$ is upper-bounded

using the partition that makes $V(f,P)$ upper-bounded in which I failed.

(in case $f$ is not continuous which is countable..)

Showing by subtraction of two increasing functions also seems hard.

Hope if someone can help me or give me a hint.

Thank you.

Answer 1

Hint: Say $f(x)\ge c>0$ for all $x$. Then $$\left|\frac1{f(x)}-\frac1{f(y)}\right|=\left|\frac{f(y)-f(x)}{f(x)f(y)}\right|\le\frac{|f(x)-f(y)|}{c^2}.$$