Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
  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?

share|improve this question
add comment

1 Answer

up vote 6 down vote accepted

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.

share|improve this answer
    
sorry I do not understand #2 @ncmathsadist –  Bunuelian Trick 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 :).... –  Bunuelian Trick 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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.