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Updated Oct 8

Problem 907

claims/: The 1 claim page of Problem 907, one per claimant's result; the problem's standing derives from them.

Statement. 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. $h(x+y)=h(x)+h(y)$ )?

Status. The site labels the problem PROVED (LEAN), and its commentary credits de Bruijn's 1951 theorem for the affirmative answer; the formal-conjectures statement file's formal_proof attribute points to a Lean proof of the statement in Alexeev's repository. Both are on the claim page. The Lean proof is not built here.

Source. erdosproblems.com/907, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #907, https://www.erdosproblems.com/907.

References.

[dB51] de Bruijn, N. G., Functions whose differences belong to a given class. Nieuw Arch. Wiskunde (2) 23 (1951), 194-218.

Formalization. Statement in formal-conjectures, whose formal_proof attribute at the pinned commit points to the proof in Alexeev's lean-proofs repository linked from the claim page; neither is built or audited here.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (7)

De Bruijn's continuous-plus-additive decomposition problems/analysis/E0907/claims Analysis analysis/debruijn_1951_functions_whose_differences_belong_given_class Theorem 1.1 (p. 194): a real function with every difference continuous is continuous plus additive Theorem 1.2 (p. 196): |f(x+y)-f(x)-f(y)+f(0)| ≤ 1 gives an additive H with |f(x)-f(0)-H(x)| ≤ 1 Theorem 4.1 (p. 204): ‖Δ_hΔ_k f‖ ≤ 1 in a Banach function space gives an additive H with ‖Δ_h(f−H)‖ ≤ 1

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