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CN functions

Source: Iran 2005

September 1, 2005
functiontrigonometrygeometry3D geometrysphereanalytic geometrytopology

Problem Statement

We call the set A∈RnA\in \mathbb R^n CN if and only if for every continuous f:A→Af:A\to A there exists some x∈Ax\in A such that f(x)=xf(x)=x. a) Example: We know that A={x∈Rn∣∣x∣≤1}A = \{ x\in\mathbb R^n | |x|\leq 1 \} is CN. b) The circle is not CN.
Which one of these sets are CN?
1) A={x∈R3∣∣x∣=1}A=\{x\in\mathbb R^3| |x|=1\}
2) The cross {(x,y)∈R2∣xy=0, ∣x∣+∣y∣≤1}\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}
3) Graph of the function f:[0,1]→Rf:[0,1]\to \mathbb R defined by f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0
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