Problem 907

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

Source: erdosproblems.com/907

Analysis

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

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 (Debruijn, 1951). The Lean proof is not built here.

Problem 907

Status

Proof claims

Discussion