Mapping the unit circle in C to R with two diametrically opposite points having the same image. I'm stuck on this problem, along with many others, from one of my Analysis problem sheets. I'd be very grateful if someone could point me in the right direction. This is from a first course in real analysis. The unit circle in $\mathbb C$ is mapped to $R$ by a map $e^{i\theta}\mapsto f(\theta)$, with $f:[0,2\pi] \to \mathbb R$ continuous and $f(0)=f(2\pi)$. Show that there exists two diametrically opposite points with the same image.

Consider the function $g(x) = f(x)-f(x+\pi)$. What can you say about $f(0)$ and $f(\pi)$? Then remember the Intermediate Value Theorem.