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Source. Laczkovich (1984), Theorem 1, printed p. 109, and its deduction from Theorem 2 on p. 110 (PDF pp. 1–2).

Statement

If $f:\mathbb R\to\mathbb R$ satisfies

$$2f(x)\le f(x+h)+f(x+2h) \qquad(x\in\mathbb R,\ h>0),$$

then $f$ is nondecreasing. No measurability, continuity, or local boundedness hypothesis is required.

Dependencies. Theorem 2 and the bounded continued fraction of $\sqrt2$ recorded in the classical input page. The preceding source-specific proof chain is part of this compilation; general continued-fraction theory is the stated external input.

Bears on. This is the affirmative real-function conclusion in Problem 1125. It does not assert strict increase or a theorem on arbitrary rational-domain functions.

Proof

Choose any real $a<b$ and define

$$g(t)=f(a+(b-a)t).$$

For every real $t$ and $s>0$, the original inequality applies with base point $a+(b-a)t$ and positive step $(b-a)s$. Hence $g$ satisfies the same inequality. Restrict $g$ to $G_{\sqrt2}=\mathbb Z\sqrt2+\mathbb Z$ and apply Theorem 2. Since $0,1\in G_{\sqrt2}$,

$$f(a)=g(0)\le g(1)=f(b).$$

This proves nondecreasing monotonicity on all of $\mathbb R$.