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=== Functions of several variables ===
* A function that is marginally convex in each individual variable is not necessarily (jointly) convex. For example, the function <math>f(x, y) = x y</math> is [[bilinear map|marginally linear]], and thus marginally convex, in each variable, but not (jointly) convex.
* A function <math>f : X \to [-\infty, \infty]</math> valued in the [[extended real numbers]] <math>[-\infty, \infty] = \R \cup \{\pm\infty\}</math> is convex if and only if its [[Epigraph (mathematics)|epigraph]] <math display=block>\{(x, r) \in X \times \R ~:~ r \geq f(x)\}</math> is a convex set.
* A differentiable function <math>f</math> defined on a convex ___domain is convex if and only if <math>f(x) \geq f(y) + \nabla f(y)^T \cdot (x-y)</math> holds for all <math>x, y</math> in the ___domain.
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