Debruijn 1951 functions whose differences belong given class

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conjecture_p195: Erdős's conjecture on functions with measurable differences as de Bruijn records it, with Erdős's remark that the two-summand decomposition fails under the continuum hypothesis.

theorem_1_1: De Bruijn's proof of Erdős's conjecture that a real function on the line whose every difference f(x+h)-f(x) is continuous in x is the sum of a continuous function and an additive function.

theorem_1_2: De Bruijn's stability theorem for the Cauchy equation on the real line, with the same constant 1 in hypothesis and conclusion, which is the key step of his first method.

theorem_4_1: De Bruijn's extension of Theorem 1.2 to a Banach space of real functions on the line, with the remarks carrying it to abelian groups and linear-space values.

theorem_5_1: De Bruijn's weak decomposition for functions whose differences are all square integrable on every finite interval, the case of Erdős's measurable-difference conjecture that the paper proves.

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

De Bruijn proves a conjecture of Erdos (Theorem 1.1): if f is real on the line and for every h the difference f(x+h)-f(x) is continuous in x, then f(x)=g(x)+H(x) with g continuous and H additive; this immediately yields the Boas-Boas theorem via Ostrowski's theorem. He formalizes the 'difference property' for a class C and shows it holds for the classes C_0 (continuous), C_2(k) (k times differentiable), C_3 (analytic), C_4 (polynomials), C_5 (absolutely continuous on finite intervals) and C_7 (bounded variation on finite intervals), while it fails for the functions bounded on the line (C_8, with the explicit example log(x^2+1)) and, by a Hamel-basis construction, for the functions bounded on every finite interval (C_9), and cannot be proved for measurable functions (Sierpinski's CH construction of a non-measurable S with countable difference sets, a remark due to Erdos). A key tool is Theorem 1.2: if |f(x+y)-f(x)-f(y)+f(0)| <= 1 for all x,y then f-f(0) is within 1 of an additive function; Theorem 4.1 extends this to a Banach space of functions, Remark 2 after it to functions on an abelian group with values in a linear space, and Theorem 4.3 gives a second, integration-based method. Erdos further conjectured that measurability of all differences gives $f=g+H+S$ with $g$ measurable, $H$ additive, and $S(x+t)=S(x)$ almost everywhere for each fixed real $t$; the exceptional null set may depend on $t$. De Bruijn states this on printed p. 195 (PDF p. 3) and proves it in Section 5, as Theorem 5.1 (printed pp. 211--212), for the functions whose every difference is square integrable on every finite interval, with $g$ then square integrable on every finite interval. This is a hypothesis on the differences, not on $f$ itself. The existing Problem 907 connection to the continuous-difference theorem is retained; the historical measurable weak-decomposition formulation is linked separately to Problem 908 below.

Source: https://www.win.tue.nl/~wsdwnb/DeBruijnPDF.html. The copy read for this card is a PDF of printed pp. 194--218, downloaded on 4 September 2026 according to its cover sheet. The file's first page is the TU/e research portal's cover sheet, which states that "Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners", that users "may download and print one copy of any publication from the public portal for the purpose of private study or research" and that "You may not further distribute the material or use it for any profit-making activity or commercial gain"; the article pages are image-only and print no notice, every other right reserved.

Read status: claims checked for Theorems 1.1 (p. 194), 1.2 (p. 196), 4.1 with its two remarks (pp. 204--205) and 5.1 (pp. 211--212), and for Erdos's remark and conjecture (p. 195); no proof checked.

Bears on. #907: Theorem 1.1 (p. 194) answers the problem's question in the affirmative; the theorem assumes every difference continuous, which the problem's hypothesis for $h>0$ implies. #908: the conjecture of p. 195 is the problem's corrected Statement, with a measurable summand, and Theorem 5.1 (pp. 211--212) proves it for the functions whose every difference is square integrable on every finite interval, a special case only.

Results. Page numbers are the printed ones (pp. 194--218).

• Theorem 1.1 (p. 194): if f(x+h)-f(x) is a continuous function of x for each h, then f=g+H with g continuous and H additive. Theorem 1.3 (p. 197) extends it to f defined on an interval.
• Theorem 1.2 (p. 196): if |f(x+y)-f(x)-f(y)+f(0)| <= 1 for all x,y, then there is an additive H with |f(x)-f(0)-H(x)| <= 1 for all x.
• Conjecture and remark (p. 195): Erdos's remark that under CH Sierpinski's nonmeasurable function has measurable differences but no decomposition $S=g+H$ with $g$ measurable and $H$ additive, and his conjecture that measurable differences give $f=g+H+S$ with $g$ measurable, $H$ additive and $S(x+h)=S(x)$ for almost all $x$, for each $h$. The remark does not refute the conjecture, whose third summand admits Sierpinski's function.
• Theorem 4.1 (p. 204): for a Banach space $\Omega$ of real functions on the line satisfying (I)--(IV), if every $\Delta_hf$ and every $\Delta_h\Delta_kf$ lies in $\Omega$ with $\|\Delta_h\Delta_kf\|\le1$, then some real-valued additive $H$ has $\Delta_h(f-H)\in\Omega$ and $\|\Delta_h(f-H)\|\le1$ for all $h$ ($f$ itself need not lie in $\Omega$). Remark 2 after it (pp. 204--205) carries it to functions on an additive abelian group with values in a linear space; with a Banach space of values and the norm $\|f(0)\|$ this gives the analogue of Theorem 1.2 in that setting, and Theorem 1.2 itself for real functions on the line.
• Theorem 5.1 (pp. 211--212): if $\Delta_hf\in C_6(2)$ for each $h$, where $C_6(2)$ is the class of functions square integrable on every finite interval, then $f=g+H+S$ with $g\in C_6(2)$, $H$ additive and, for each $h$, $S(x+h)=S(x)$ for almost all $x$.

E908 formulation scope. The 1951 measurable-summand statement differs from the continuous-summand wording in Erdős's 1982 retrospective. Laczkovich 1980 Theorem 3 proves the measurable weak difference property. The stronger essentially-continuous-difference variant is distinguished on Problem 908. The remaining results summarized in this digest (Theorems 1.3, 4.2, 4.3 and 6.1, the class C_5 of Section 5 and the classes of Sections 3, 6 and 7) are retained without new proof review. Local reinspection of p. 195: 6 September 2026 UTC.

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