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.

  • $\begingroup$ Welcome to math.se! As in TeX you can use $ for formatting of mathematics. $\endgroup$ – Smylic Apr 3 '17 at 7:33
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    $\begingroup$ It doesn't. Consider rotating the circle by 45 degrees for example $\endgroup$ – mathworker21 Apr 3 '17 at 7:34
  • $\begingroup$ If this is not the systematic outsourcing of one's homework, I don't know what is. $\endgroup$ – Did Apr 5 '17 at 7:04

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.


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