# Function of bounded variation whose reciprocal is not of bounded variation

Problem: Find an example of a positive function $$f: [0,1] \to \mathbb{R}_{>0}$$ that is of bounded variation, whose reciprocal $$1/f$$ is integrable but not of bounded variation.

One necessary condition for $$f$$ is that $$\inf_{x \in [0,1]} f(x)=0$$, but I don't know how to proceed further.

• Hint: make $1/f$ an integrable function taking all values in $\mathbb{N}-\{0\}$. – user10354138 Jul 19 at 4:33

You have more or less resolved this with the observation that $$\inf_{x \in [0,1]} f(x)=0$$. Just take the simplest such function, e.g. $$f(x)= \begin{cases}\sqrt{(x)},\qquad x\neq0,\\1,\qquad x=0.\end{cases}$$
Note that any continuous $$f$$ with the property $$\inf_{x \in [0,1]} f(x)=0$$, will not be positive.