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.
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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