Problem about continuous functions and the intermediate value theorem

The idea is good, but the intermediate value theorem applies to functions mapping a closed interval $I \subset \Bbb R$ to $\Bbb R$.

It would be possible to formulate a similar statement for functions $\phi: S^1 \to \Bbb R$, but it is simpler to consider $$ \phi: [0, \pi] \to \Bbb R, \quad \phi(\alpha) = f(\cos\alpha, \sin\alpha) - f(-\cos\alpha, -\sin\alpha) $$ instead, so that the IVT can be applied directly.

The remaining argument can also be simplified. It suffices to observe that $\phi(0) = - \phi(\pi)$, so that

• either $\phi(0) = \phi(\pi) = 0$,
• or $\phi(0)$ and $ \phi(\pi)$ have opposite sign, and the intermediate value theorem states that there is some $\alpha \in (0, \pi)$ with $\phi(\alpha) =0$.