Example of a function that has the Luzin $n$-property and is not absolutely continuous.

Question

The Banach–Zaretsky theorem (page 196) says that a continuous function $f:[a,b]\to\mathbb{R}$ of bounded variation is absolutely continuous if and only if

$$E\subset I \text{ has zero Lebesgue measure }\Rightarrow f(E) \text{ has zero Lebesgue measure }\;\;[\#]$$

I would like see an example of a function that satisfies $[\#]$ but is not absolutely continuous.

Thanks.