continuous surjective function from $n$-sphere to unit interval

Question

Let $S^n:=\{x \in \mathbb R^{n+1} : ||x||=1\}$ . Let $f : S^n \to [0,1]$ be a continuous surjective function , then is it necessarily true that $f^{-1}\{x\}$ is infinite $ \forall x \in [0,1]$ ?

Answer 1

No. Take $f(x_1,\ldots,x_{n+1})=\frac{x_1+1}2$. Then $f^{-1}(1)=\{(1,0,0,\ldots,0)\}$.

Answer 2

It is not true. Take a point $x_0\in S^n$, and consider the continuous map $f(x):=\frac{1}{2}\|x-x_0\|$. Then, for any $x\in S^n$, $$0\leq f(x)\leq \frac{\|x\|+\|x_0\|}{2}\leq 1.$$ Moreover $f(-x_0)=\|x_0\|=1$ and $f^{-1}(0)=\{x_0\}$.