Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics II / 5 / a

The intermediate value theorem states that if $f:[u,v]\to \mathbb{R}$ is continuous and $y$ lies between $f(u)$ and $f(v)$, inclusive, then some $c\in [u,v]$ satisfies $f(c)=y$. If $y$ lies strictly between the endpoint values, then $c$ lies in $(u,v)$.

For $n=1$, choose $x=0$, $y=1$. For $n\ge 2$, define the continuous functionIts values at the grid points have the telescoping sumIf one summand is zero, we already have a zero of $g$. Otherwise the summands cannot all have the same sign, so two of these values have opposite signs. The intermediate value theorem between their arguments gives $t\in [0,1-1/n]$ with $g(t)=0$. Set $x=t$, $y=t+1/n$. ThenThis is the horizontal chord of reciprocal-integer length property; the telescoping sum explains why reciprocal integers appear.

An explicit piecewise linear function isThe adjacent formulas agree at $1/3$ and $2/3$, so the function is continuous, and $f(0)=f(1)=0$. Any pair of points at distance $2/3$ can be written $t,t+2/3$ with $0\le t\le 1/3$. For every such $t$,Thus their values are never equal. The horizontal chord of reciprocal-integer length conclusion does not extend to this distance.