True/False: properties of a continuous map $f: S^1 \to \Bbb R$.

Question

Let $S^1=\{z \in \Bbb C : |z|=1\}$. Let $f$ be any continuous function from $S^1 \to \Bbb R$. Then:

A. $f$ is bounded

B. $f$ is uniformly continuous

C. $f$ has image containing a non-empty open subset of $\Bbb R$

D. $f$ has a point $z \in S^1$ such that $f(z)=f(-z)$.

Since $S^1$ is compact and connected, we get $\text {Image}(f)$ to be a closed & bounded interval so option A & C are true.

Option B seems to be true because $f$ is continuous on a compact set $S^1$.

Also option D is just Borsuk-Ulam theorem.

Where have I gone wrong?

Answer 1

C is wrong as $f$ might be constant.

Answer 2

C is true if $f$ is nontrivial, for then $f[S^1]$ is a nontrivial connected subset of $\mathbb R$.