Corollary 1

Tag 087P, permanent. Cite it as [Bourbaki, Tag 087P].

Functions of a Real Variable, chapter II, § 1. PRIMITIVES AND INTEGRALS

If a real regulated function $f$ on I is such that $f(x) \geq 0$ at the points where $f$ is continuous, then $\frac{1}{b-a} \int_a^b f(t)\,dt > 0$ unless $f(x) = 0$ at the points where $f$ is continuous.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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