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Functions for which fof=[id] are not short maps, nor contractions

Source: Romanian District Olympiad 2011, Grade IX, Problem 3

October 8, 2018
functionalgebraInteger PartFloor

Problem Statement

Let f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} be a function with the property that (f∘f)(x)=[x], (f\circ f) (x) =[x], for any real number x. x. Show that there exist two distinct real numbers a,b a,b so that ∣f(a)−f(b)∣≥∣a−b∣. |f(a)-f(b)|\ge |a-b|.
[] [] denotes the integer part.
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