Brouwer's fixed point theorem

Question

Is the statement is true or false, please explain the reason
Every continuous map $f \colon S^1 \to S^1$ has a fixed point where $S^1$ is a unit circle in $\mathbb R^2$ follows from Brouwer's fixed point theorem.

Answer 1

Brouwer talks about the compact disk. You're asking about the boundary of the disk. Imagine the disk as a basin of water, where the continuous function is "swirl around the centre". Then Brouwer guarantees a fixed-point across the entire basin (here, the centre is fixed), but not the edge.