lean_workbook_plus_189

Proved

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

Prove that if $f(av)=af(v)$ and $f(v+w)=f(v)+f(w)$ for all $a, v, w \in \mathbb{R}$ , then $f$ is a linear function of the form $f(x) = cx$ for some constant $c$ .

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_189 (f : ℝ → ℝ) (hf: ∀ a v w : ℝ, f (a * v) = a * f v ∧ f (v + w) = f v + f w) : ∃ c :ℝ, ∀ x : ℝ, f x = c * x   :=  by sorry

Source

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

View graph