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.