Bourbaki

Exercise 18, § 1

Functions of a Real Variable, II, 83

Let ff be a finite real function, continuous on an open interval II. For ff to be convex on II 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
(argue as in exerc. 9 of I, p. 46).

Solution

No solution has been written yet.

Transcribed from the printed volume. Not translated.