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Let $S=\{x\in \mathbb R^n: \|x\|=1\}$ be the unit sphere in $\mathbb R^n$, and let $f: S\to \mathbb R$ be a real-valued continuous function on $S$. Prove that there is a point a belonging to $S$ such that $f(a)=f(-a)$.

Cory
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  • I suggest you take a look at (http://en.wikipedia.org/wiki/Intermediate_value_theorem#Implications_of_theorem_in_real_world) – Pedro Oct 01 '14 at 17:59

1 Answers1

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Suppose, for the sake of a contradiction, there is no such $a\in S$. Define the function $g:S\to \mathbb{R}$ by $g(x)=f(x)-f(-x)$. Our assumption guarantees that $g$ is never zero. By the intermediate value theorem, this implies that $g$ is either always positive or always negative. Without loss of generality assume that $g(x)>0$ for every $x\in S$. But then for any point $x\in S$, we have $0<g(-x)=f(-x)-f(x)=-g(x)<0$, a contradiction.

Adam Azzam
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    So you're saying there is no point ? – Cory Oct 01 '14 at 18:31
  • There is such a point. There was a small typo in my first sentence which is corrected. I assumed there is no such point, and reached a contradiction. Hope this helps. – Adam Azzam Oct 01 '14 at 18:45