lean_workbook_plus_18879

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

Find all the continuous functions $f: [0,\infty) - > [0,\infty)$ such that $f(f(x)) = \sqrt {xf(x)}, \forall x \in [0,\infty)$

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_18879 (f : ℝ → ℝ) (hf: Continuous f) (h: f '' Set.Ici 0 ⊆ Set.Ici 0)(hf2: ∀ x ∈ Set.Ici 0, f (f x) = Real.sqrt (x * f x)) : ∃ g : ℝ → ℝ, Continuous g ∧ g '' Set.Ici 0 ⊆ Set.Ici 0 ∧ (∀ x ∈ Set.Ici 0, g (g x) = Real.sqrt (x * g x))   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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