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Piotr Hajlasz
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There are strictly increasing functions that map a set of positive measure to a set of measure zero. I will focus on functions defined on $[0,1]$, but the construction can be extended to $f:\mathbb{R}\to\mathbb{R}$. Here is a construction:

Let

Piotr Hajlasz
  • 28k
  • 5
  • 86
  • 186