A real function sending bounded sets to bounded sets is necessarily of bounded variation?

Question

A real function sending bounded sets to bounded sets is necessarily of bounded variation?

Answer 1

No. Take$$\begin{array}{ccc}\mathbb{R}&\longrightarrow&\mathbb{R}\\x&\mapsto&\begin{cases}\sin\left(\frac1x\right)&\text{ if }x\neq0\\0&\text{ otherwise}\end{cases}\end{array}$$maps any set into a bounded set, but is not of bounded variation.