# Continous function with measure of image of zero set positive

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?

Best KingDingeling

• Indeed something with the Cantor set helps you, since it is uncountable and has measure zero. This might help you math.stackexchange.com/q/40504/349501 Commented Dec 1, 2018 at 23:25
• This link definitely helped, thank you very much. Commented Dec 2, 2018 at 12:30

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 $$1$$.
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.