# Convexity/Convex functions

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 ${\displaystyle \displaystyle 0\leq \lambda \leq 1,}$

${\displaystyle \displaystyle f(\lambda x+(1-\lambda )y)\leq \lambda f(x)+(1-\lambda )f(y).}$

NB: Because X is convex, :${\displaystyle \displaystyle (\lambda x+(1-\lambda )y)}$ 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.