# Convexity/Convex functions

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A **convex function** f(**x**) is a real-valued function defined over a convex set X in a vector space such that for any two points x, y in the set and for any λ with

NB: Because X is convex, : must be in X.

If the function -f(x) is convex, f(x) is said to ba a **concave function**. It is easily seen that if a function is both convex and concave, it must be linear.

**Theorem:** A convex function on X is bounded above on any compact subset of X.

**Theorem:** A convex function on X is continuous at each point of the interior of X.

**Theorem:** If f(x) is convex in a set containing the origin O, and f(O) = 0, then ^{f(μx)}⁄_{μ} is an increasing function of μ for μ > 0.

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