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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?

Bach
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1 Answers1

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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.

Kavi Rama Murthy
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