# Is there a continuous transformation that does not preserve zero measure?

Is there a continuous transformation that does NOT preserve zero measure?

I found one of my textbooks claims that a continuous transformation in $$\mathbb R^n$$ preserves zero measure. But I am highly skeptical about it.

In fact, I can only see the case when the transformation is actually Lipschitz, like in this post. But in general, I am struggling in finding a counterexample. Any help?

The map $$\sum \frac {2a_n} {3^{n}} \to \sum \frac {a_n} {2^{n}}$$ ($$a_n \in \{0,1\}$$ for all $$n$$) is a continuous map from the Cantor set $$C$$ onto $$[0,1]$$. This map extends to a continuous function on $$\mathbb R$$ since its vales at the end points of the intervals removed in the construction of $$C$$ are equal). This extended function is called the Cantor function.