Find functions $f,g \in BV([0,1])$ with $f/g \not \in BV([0,1])$.

Question

Find functions $f,g \in BV([0,1])$ such that $() > 0$ on $[0, 1]$ and $/ ∉ ([0, 1]).$ I was trying something like $f=1$ and $g=x$ etc. but there is always a problem with $0$.

Answer 1

If you are interested in an example where $f,g$ and $f/g$ are all bounded, you might try

$$f(x)= \begin{cases} x^2\cdot\sin(1/x), & x\in(0,1],\\ 0, & x=0, \end{cases}$$

and

$$g(x)=\begin{cases} x, & x\in(0,1],\\ 1, & x=0. \end{cases}$$

You may note that $f$ and $f/g$ are even continuous.

Answer 2

You can use $f \equiv 1$ and

$$g(x)= \begin{cases} 1 & \text{if $x=0$}\\ x & \text{if $x \in (0,1]$} \end{cases}$$

(The point is to use a function $g$ that is never $0$, but arbitrarily approaches $0$.)