Midpointconvexity

Midpoint convexity is a property of certain functions. A function f: I -> R, where I is an interval of real numbers, is said to be midpoint convex if for any two points x and y in I, the value of the function at the midpoint of x and y is less than or equal to the average of the function values at x and y. Mathematically, this can be expressed as f((x+y)/2) <= (f(x) + f(y))/2 for all x, y in I.

This definition is closely related to the concept of convexity. In fact, any convex function is also midpoint convex. However, the converse is not always true; there exist functions that are midpoint convex but not convex. These non-convex midpoint convex functions are often described as "pathological" and are typically discontinuous.

For continuous functions, midpoint convexity is equivalent to convexity. This means that if a function is both continuous and midpoint convex, then it must also be convex. This equivalence is a significant result in the study of functional equations and analysis.

The study of midpoint convexity arises in various areas of mathematics, including functional analysis, optimization theory, and the theory of differential equations. Understanding the distinction between midpoint convexity and full convexity is important for characterizing the behavior of functions and solving problems where these properties are relevant.

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