An Impossible Half-Shift?

★ 8/10functions number theory algebra Induction Functional Equations

Problem

A real function $f$ satisfies

f(f(x)) = x - 1 \quad \text{for all real } x.

Do there exist integers $n$ with $f(n)$ also an integer? Find all such $n$ or prove none exist.

Answer

Solution

Difficulty	8/10
Topics	functions, number theory, algebra, Induction, Functional Equations

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