Can any one please give a hint for finding the solution of the following question? I can't imagine such a function should exist.
Definition: Let $S$ be a nonempty subset of a linear space. We say that a map $\psi\in \mathbb{R}^S$ is pseudo-affine if $\psi (\lambda x+(1-\lambda)y)=\lambda \psi(x)+(1-\lambda)\psi(y)$ for all $0\leq \lambda\leq 1$ and $x,y\in S$ with $\lambda x+(1-\lambda)y \in S$
(a) Give an example of a nonempty subset $S\in \mathbb{R}^2$ and a pseudo-affine $\psi\in \mathbb{R}^S$ such that is not affine.
Here, an affine function is defined by the following.
Definition: A real map $\psi\in\mathbb{R}^S$ is called affine if
$\psi(\sum_{x\in A}\lambda(x)x)=\sum \lambda(x)\psi(x)$
for any $A\in P(S)$ (the class of all nonempty finite subsets of $S$), and $\lambda\in\mathbb{R}^A$ such that $\sum \lambda(x)=1$ and $\sum \lambda(x)x\in S$