Proposition 5

Tag 085Z, permanent. Cite it as [Bourbaki, Tag 085Z].

Functions of a Real Variable, chapter I, § 4. CONVEX FUNCTIONS OF A REAL VARIABLE

For a real finite function $f$ to be convex (resp. strictly convex) on an interval $I$ it is necessary and sufficient that for all $a \in I$ the gradient

p(M_a M_x) = \frac{f(x) - f(a)}{x - a}

be an increasing (resp. strictly increasing) function of $x$ on $I \cap \mathbf{C}\{a\}$ .

This proposition is an immediate consequence of definitions 1 and 2 and of the lemma of I, p. 23.

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

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