Jensen's functional equation

Remarks

Example: $f(x) = 2x + 3$ is a solution, since both sides equal $x + y + 3$. Non-example: $f(x) = x^2$ is not a solution, since the right side minus the left side is $(x - y)^2 / 4$, which is positive for $x \ne y$. If $f$ is a solution, then $A(x) = f(x) - f(0)$ satisfies $A(x/2) = A(x)/2$ and then $A(x + y) = A(x) + A(y)$. So the solutions are $f = A + b$ with $A$ an additive function and $b \in \R$. The same equation makes sense for functions on a real interval or on a convex subset of a vector space.