I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$.
The piecewise continuity requirement can be replaced by the following two other conditions: that $f$ is integrable, and there exists positive $K$ such that $|f(x)-f(y)|\geq K|x-y|$, i.e. something like the opposite of Lipschitz condition.
Aleph thanks!