Remark

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

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

On sees in the same way that the set of points lying strictly above G is convex. Conversely, if this set is convex one has

\lambda y + (1 - \lambda)y' > f(\lambda x + (1 - \lambda)x')

for $0 \leq \lambda \leq 1$ and $y > f(x),\ y' > f(x')$ ; on letting y tend to $f(x)$ and $y'$ approach $f(x')$ in this formula it follows that $f$ is convex.

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

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Cites

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