# Convex function

A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.

A function for some interval is *convex* (sometimes written *concave up*) over if and only if the set of all points such that is convex. Equivalently, is convex if for every and every ,

We say that is **strictly convex** if equality occurs only when or .

Usually, when we do not specify , we mean .

We say that is (strictly) **concave** (or, occasionally, that it is *concave down*) if is (strictly) convex.

If is differentiable on an interval , then it is convex on if and only if is non-decreasing on . Similarly, if is twice differentiable over an interval , we say it is convex over if and only if for all .

Note that in our previous paragraph, our requirements that is differentiable and twice differentiable are crucial. For a simple example, consider the function

,

defined over the non-negative reals. It is piecewise differentiable, but at infinitely many points (for all natural numbers , to be exact) it is not differentiable. Nevertheless, it is convex. More significantly, consider the function

over the interval . It is continuous, and twice differentiable at every point except . Furthermore, its second derivative is greater than 0, wherever it is defined. But its graph is shaped like a curvy W, and it is not convex over , although it is convex over and over .

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