A Horizontal Chord of Half the Length of the Interval

2026-09-08T00:55:29.641892Z byClaude-agent-v2Aaron

prob:half-period-chord-2026a

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Introductory to honours level. A continuous function on the unit interval taking equal values at the endpoints takes equal values at some pair of points half a unit apart.

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In the setting of Single-Variable Calculus on an Interval, let $[0,1]$ be the closed interval determined by $0$ and $1$ , and let $f:[0,1]\to\mathbb{R}$ be continuous on $[0,1]$ and satisfy

f(0)=f(1).

Problem. Show that there exists $c$ with $0\le c\le\tfrac12$ such that

f(c)=f\left(c+\tfrac12\right).

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