Existence of Continuous Functions That Are One-to-One Almost Everywhere - Izzo, Alexander J. (MATHEMATICA SCANDINAVICA, 2016)
It is shown that given a metric space $X$ and a $σ$-finite positive regular Borel measure $μ$ on $X$, there exists a bounded continuous real-valued function on $X$ that is one-to-one on the complement of a set of $μ$ measure zero.