Problem 908

Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is measurable for every $h>0$ . Is it true that

f=g+h+r

where $g$ is continuous, $h$ is additive (so $h(x+y)=h(x)+h(y)$ ), and $r(x+h)-r(x)=0$ for every $h$ and almost all (depending on $h$ ) $x$ ?

Second statement

Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is measurable for every $h>0$ . Is it true that

f=g+h+r

where $g$ is measurable, $h$ is additive (so $h(x+y)=h(x)+h(y)$ ), and $r(x+h)-r(x)=0$ for every $h$ and almost all (depending on $h$ ) $x$ ?

Source: erdosproblems.com/908

Analysis

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED, the site's label (page last edited 30 December 2025), which describes the corrected Statement: the site's commentary calls the problem a conjecture of de Bruijn and Erdős and credits Laczkovich [La80] with the affirmative answer. Laczkovich's Theorem 3 proves the corrected Statement and is recorded as an accepted full claim.

Notes

The change replaces "continuous" by "measurable" as the condition on $g$ ; nothing else changes. The evidence is the posers' own words. De Bruijn, whom the site credits as co-poser, records the conjecture as Erdős's in [dB51], printed p. 195, "where $g(x)$ is measurable". Erdős's 1982 retrospective [Er82e], Chapter V, §3, printed p. 76, prints "where $g(x)$ is continuous", cites [dB51] and [La80] beside the problem and reports that Laczkovich proved it. Laczkovich proved the measurable form: his Theorem 3 [La80], printed p. 224, proves it, and his introduction, printed p. 217, states Erdős's conjecture with $g$ measurable. The defect is already in [Er82e], and the site's wording follows it. The site's label and the problem's standing judge the corrected Statement. The site also cites [Er81b], Erdős's 'Problems' in The Scottish Book (1981), whose statement of the problem is not recorded here; the correction rests on [dB51], [Er82e] and [La80].

La80: Laczkovich, M., Functions with measurable differences. Acta Math. Acad. Sci. Hungar. (1980), 217-235. Introduction, printed p. 217; Theorem 3, printed p. 224. Library home: Laczkovich 1980.
dB51: de Bruijn, N. G., Functions whose differences belong to a given class. Nieuw Arch. Wiskunde (2) 23 (1951), 194--218. Erdős's conjecture with a measurable summand, printed p. 195. Library home: Debruijn 1951.
Er81b: Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27-35 (page numbers are given for the 2nd edition of The Scottish Book).
Er82e: Erdős, P., Some of my favourite problems which recently have been solved. Proceedings of the International Mathematical Conference (Singapore, 1981), North-Holland Math. Stud. 74 (1982), 59--79. Chapter V, §3, printed p. 76. Library home: Erdős 1982.
Ke98: Keleti, T., Difference functions of periodic measurable functions. Fund. Math. 157 (1998), 15--32. Theorems 2.9 and 2.13, printed pp. 21--22; Corollary 2.18, printed p. 23. Library home: Keleti 1998.

Problem 908

Second statement

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Notes

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Discussion