FormalConjectures.ErdosProblems.«907»

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import FormalConjecturesUtil

Erdős Problem 907

References:

erdosproblems.com/907

[dB51] de Bruijn, N. G., Functions whose differences belong to a given class. Nieuw Arch. Wiskunde (2) (1951), 194-218.

namespace Erdos907

Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$. Is it true that $$f=g+h$$ for some continuous $g$ and additive $h$ (i.e. $h(x+y)=h(x)+h(y)$)?

A conjecture of Erdős from the early 1950s. Answered in the affirmative by de Bruijn [dB51].

@[category research solved, AMS 26 39, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos907.lean"]
theorem erdos_907 : answer(True) ↔
    ∀ f : ℝ → ℝ, (∀ h : ℝ, 0 < h → Continuous fun x => f (x + h) - f x) →
      ∃ g a : ℝ → ℝ, Continuous g ∧ (∀ x y, a (x + y) = a x + a y) ∧ f = g + a := by⊢ True ↔
  ∀ (f : ℝ → ℝ),
    (∀ (h : ℝ), 0 < h → Continuous fun x ↦ f (x + h) - f x) →
      ∃ g a, Continuous g ∧ (∀ (x y : ℝ), a (x + y) = a x + a y) ∧ f = g + a
  sorryAll goals completed! 🐙

end Erdos907