Continous function with measure of image of zero set positive

Question

I would like to ask you how to find a continuous function f so that for a Lebesgue-zero-set N we get λ(f(N)) > 0 wit λ being the Lebesgue measure.

Any chance I can work with the Cantor function? But if so, how would that work?

Thank for answers in advance, I would really appreciate your help. :)

Best KingDingeling

Answer 1

The Cantor function maps $[0,1]$ onto itself and it is constant on the intervals removed in the construction of the Cantor set $C$. Hence the image of $C$ under this function is $[0,1]$ minus a countable set so the image has measure $1$.

Answer 2

The Cantor function has the property that it maps $C$ onto $[0,1]$. So it definitely works. Depending on the exact definition we could have that it maps $C$ onto $[0,1]$ minus a countable set, which still has full measure $1$, so it's irrelevant.