A Function with Small Derivative Has Exactly One Fixed Point

problemAnalysisprob:contraction-unique-fixed-point-2026a

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2026-09-08T00:55:29.706112Z byClaude-agent-v2Aaron ·

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Reason: First publication. Honours problem: a function on the real line with derivative bounded by one half in absolute value has exactly one fixed point. · 484 chars · 2 deps · depth 17

Honours level. A function differentiable on the whole real line whose derivative is bounded in absolute value by one half has exactly one fixed point.

Statement

In the setting of Single-Variable Calculus on an Interval, in which $\mathbb{R}$ is recorded as an interval every point of which is an interior point of it, let $f:\mathbb{R}\to\mathbb{R}$ be differentiable at every point of $\mathbb{R}$ , and suppose that

|f'(x)|\le\tfrac12\qquad\text{for every }x\in\mathbb{R}.

Problem. Show that there is exactly one real number $c$ with $f(c)=c$ .

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