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Is $|f^{-1}f(p)|$ constant on a conull set?
Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is an analytic map. Can we say that there is a conull set $U \subset \mathbb{R}^n$ (i.e. $\mathbb{R}^n \setminus U$ has measure zero) where $|f^{-1}f …