A continuous map from $D$ unit disk, to $S^1$

1. $f:D\rightarrow S^1$ is a continuous then $\exists x\in S^1$ such that $f(x)=x$?

2. $f:S^1\rightarrow S^1$ then same as 1 holdd?

3. $f:E\rightarrow E$ then same as 1 hold? $E=\{(x,y):2x^2+3y^2\le 1\}$

by Fixed point Theorem I know 2,3 are correct, what about 1?

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1 Answer

In #1, you can think of $f: D\rightarrow D$. The fixed-point theorem says that there exists $x\in D$ so that $f(x) = x$.

In #2, any rotation that is not a complete rotation has no fixed point.

In #3, $E$ is homeomorphic to the unit disk. It must have a fixed point.

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sorry I do not understand #2 @ncmathsadist –  La Belle Noiseuse Jul 12 '12 at 16:41
Simple example $f: S^1 \to S^1$, $f(z) = -z$, has no fixed points. –  Justin Young Jul 12 '12 at 17:35
Thank you :).... –  La Belle Noiseuse Jul 12 '12 at 18:36
By that I mean any rotation that is not a multiple of $2\pi$. –  ncmathsadist Jul 12 '12 at 22:41