 Jensen functional equation

The Jensen functional equation on $\mathbb{R}$ isIt says that a function preserves midpoint averages. Every affine function $f(x)=ax+b$ satisfies it. The quadratic function $x^2$ does not: the midpoint of $1,-1$ is zero but the average of their squares is one.

Set $g(x)=f(x)-f(0)$. Taking $y=0$ gives $g(x/2)=g(x)/2$, and substituting this back into the equation yields $g(x+y)=g(x)+g(y)$. Thus solutions are exactly a constant plus a solution of the Cauchy functional equation. Continuity at one point forces $g(x)=ax$, first on rational multiples by additivity and then on all real inputs by rational approximation. Without a regularity hypothesis, non-linear additive solutions can occur, so the midpoint identity alone is not sufficient to infer an affine formula.