[Math] Example of a set $Y$ that has zero Lebesgue measure and a continuous function $f$ such that $f(Y)$ is not a set of zero Lebesgue measure.

Question

Could someone give me an example of a set $Y\subset \mathbb{R}$ that has zero Lebesgue measure and a continuous function $f:X\subset \mathbb{R}\to\mathbb{R}$ such that $Y\subset X$ and $f(Y)$ is not a set of zero Lebesgue measure?

Thanks.

Answer 1

The devils staircase on the Cantor set. The Cantor set is a null set and its image is $[0,1]$

The Cantor set can be written as all $x$ of $\mathbb{R}$ such that $$x=\sum_{n=1}^\infty \frac{a_n}{3^n}$$ where $a_n\in\{0,2\}$ and your functions maps $x$ to $$f(x)=\sum_{n=1}^\infty \frac{a_n}{2^{n+1}}$$ when $x$ is in the Cantor set and the trivial extension on $[0,1]$ (It is constant outside the Cantor set).