Jensen's functional equation

Summary

Jensen's functional equation is the functional equation $f\left(\frac{x + y}{2}\right) = \frac{f(x) + f(y)}{2}$ for an unknown function $f \colon \R \to \R$. It says that the value at the midpoint of two points is the average of the values. Its solutions are the functions $f = A + b$ with $A$ an additive function and $b$ a constant, and the continuous ones are the affine functions $f(x) = ax + b$. It is the equality case of midpoint convexity, which links it to convex functions.