ca.analysis and odes - Existence of an "anti-additive" (or "never linear") map?

(I've edited this question)

I'm searching for a continuously differentiable function $f:mathbb R^2tomathbb R$ 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 $f:mathbb R^2tomathbb R$. 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.)