(I've edited this question)

I'm searching for a continuously differentiable function $f:\mathbb R^2\to\mathbb R$ such that $f(x)+f(x+u+v)\neq f(x+u)+f(x+v)$ for all $x$ and all linearly independent $u$ and $v$.

My original question was about the special case $x=0, f(x)=0$ for merely continuous functions, which turned out to be trivial.

(I was lead to this question when investigating whether one can always find the vertices of a parallelogram (or more specifically, a square) in the graph of a continuously differentiable function $f:\mathbb R^2\to\mathbb R$. The nonexistence of functions such as the above would imply that one cannot always find a parallelogram in the graph of a continuously differentiable function.)

existinganswers to it stop being answers. If you want to ask another question, ask another question! – Mariano Suárez-Alvarez♦ Feb 8 '10 at 19:25