$f$ continuous on a circle, then there is a diameter at which ends $f$ has the same value Let $f$ be a continuous function on a circle.
Show that there is a diameter for which the value of $f$ are equal at the diameter ends.
I don't know how to approah this problem.
Is a diameter any chord of the circle?
I tried to consider $f(\theta) = e^{i\theta}$ which satisfies the condition for the chord $y=f(\pi/4)$.
I don't see how to extend this to a general continuous function.

A diameter is a chord that passes through the centre of the circle. 
HINT: For $0\le\theta\le\pi$ let $$g(\theta)=f\big(e^{i\theta}\big)-f\big(e^{i(\theta+\pi)}\big)\;,$$ the difference in the values of $f$ at the opposite ends of a diameter. Use the intermediate value theorem.
