Paper 1, Section II, $11 D$

(i) State and prove the intermediate value theorem.

(ii) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. The chord joining the points $(\alpha, f(\alpha))$ and $(\beta, f(\beta))$ of the curve $y=f(x)$ is said to be horizontal if $f(\alpha)=f(\beta)$. Suppose that the chord joining the points $(0, f(0))$ and $(1, f(1))$ is horizontal. By considering the function $g$ defined on $\left[0, \frac{1}{2}\right]$ by

$g(x)=f\left(x+\frac{1}{2}\right)-f(x)$

or otherwise, show that the curve $y=f(x)$ has a horizontal chord of length $\frac{1}{2}$ in $[0,1]$. Show, more generally, that it has a horizontal chord of length $\frac{1}{n}$ for each positive integer $n$.

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