Bourbaki

Exercise 18, § 1

Functions of a Real Variable, II, p. 32

Let f be a finite numerical function continuous in an open interval I. In order that f be convex in I, it is necessary and sufficient that, for every x∈Ix \in I, one has

lim sup⁡h→∞(12h∫x−hx+hf(t) dt−f(x))≥0 \limsup_{h \to \infty} \left( \frac{1}{2h} \int_{x-h}^{x+h} f(t)\,dt - f(x) \right) \geq 0
(reason as in Exercise 9 of I, p. 52).

Solution

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Machine translation of the English, by gpt-5-6-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.