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1: Let $f\colon S^1 \to \mathbb{R}$ be any continuous map, where $S^1$ is the unit circle in the plane. Let:

$$A = \{(x, y) \in S^1 \times S^1 : x\ne y, f(x) = f(y)\}$$

Is $A$ non-empty? If the answer is ‘yes’, is it finite, countable or uncountable?

2: Let $f\colon S^1 \to \mathbb{R}$ be any continuous map, where $S^1$ is the unit circle in the plane. Let:

$$A = \{(x, y)\in S^1\times S^1: x = −y, f(x) = f(y)\}$$

Is A non-empty?

Please help anybody. I have no idea.

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For 2: A is not always empty. take $f(x)=c$ for all $x$. Same idea for 1. –  Lucien Sep 10 '12 at 13:43
For 1) observe that $f$ attains its maximum and its minimum at two points $x_0,x_1 \in S^1$. This divides $S^1$ into two intervals. What can you say about their images? –  t.b. Sep 10 '12 at 13:43
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1 Answer

up vote 2 down vote accepted

We can identify $S^1$ with the interval $[0,1)$ via $e^{i\theta} \mapsto \dfrac{\theta}{2\pi} \pmod 1$, and so we can consider $f$ as a continuous function $[0,1] \to \mathbb{R}$ with $f(0)=f(1)$. We know that $f$ has to be bounded and attain its bounds, so let $x_*, x^* \in [0,1]$ with $f(x_*) \le f(x) \le f(x^*)$ for all $x \in [0,1]$.

Suppose $f(x^*) > f(0)=f(1)$. Then in particular $0<x^*<1$. By the intermediate value theorem, for each $0<t<1$, there exist $\xi_t \in (0,x^*)$ and $\eta_t \in (x^*,1)$ with $f(\xi_t) = t \cdot f(x^*) = f(\eta_t)$, and so $(\xi_t, \eta_t) \in A$. I'll let you consider the other cases; this answers (1).

For (2), consider the function $g : S^1 \to \mathbb{R}$ given by $g(x)=f(x)-f(-x)$. Prove that either $g \equiv 0$ or $g$ takes both positive and negative values. Hint: express $g(-x)$ in terms of $g(x)$.

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Not leaving much for the OP to do... –  t.b. Sep 10 '12 at 13:51
@t.b.: I may have mis-stricken the balance between 'being helpful' and 'doing their homework', but it wasn't tagged as homework. There's still plenty to do (e.g. the other cases in (1), the countability or otherwise of $A$ in (2)). –  Clive Newstead Sep 10 '12 at 13:53
Sure, I understand. It's just that this particular OP posts one such question after the other and the only contribution is "i think ..." or "i have no idea ..." I don't know if it's homework, but it sure has the distinct smell of it. Also, before the edit this question looked like this... –  t.b. Sep 10 '12 at 13:54
@t.b.: Fair enough, I'll keep this in mind for future reference. –  Clive Newstead Sep 10 '12 at 13:56
This exercice (2) admits an interesting generalization, naimly that for any continuous function $f:S^n\to\mathbb{R}$ ($n\geq1$), there exist antipodal points, i.e. such that $f(x)=f(-x)$. The proof is exactly the same. –  Lucien Sep 10 '12 at 13:58
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