Jensen's functional equation

Jensen's functional equation involves a function \(f\) satisfying the condition \(f\left(\frac{x + y}{2}\right) = \frac{f(x) + f(y)}{2}\). This means that applying \(f\) to the midpoint between two points yields the same result as averaging the values of \(f\) at those points. It characterizes functions that preserve midpoints, often leading to the conclusion that such functions are linear or affine (straight-line) under certain conditions. In essence, Jensen's equation captures the idea of functions that are consistent with the concept of midpoint or average behavior, reflecting a form of symmetry and linearity.