Continuous function $f \colon S^n \to [0,1]$ that is not concentrated

Question

I would like to construct a continuous function from a sphere to the unit interval $f\colon S^n \to [0,1]$ that is not concentrated. In other words, for every $n$ and every $x\in[0,1]$, the set $f^{-1}(x-\frac{1}{3},x+\frac{1}{3})$ has measure at most $\frac{2}{3}$.

Any advice on this please?

Answer 1

Assuming the measure on $S^n$ is $\sigma$, normalized so $\sigma(S^n)=1$:

First say $\phi(\xi)=\xi_1$ (the first coordinate of $\xi$). Think about $$f(\xi)=\sigma\left(\{\xi'\in S^n:\phi(\xi')<\phi(\xi)\}\right).$$