Is there a fucntion such that:
$$F: \mathbb{N}^2 \rightarrow \mathbb{N}$$ $$C_1: \mathbb{N}^4 \rightarrow \mathbb{N},\, C_2: \mathbb{N}^4 \rightarrow \mathbb{N}$$ $$ \left. \begin{matrix} x_1 - a_x < x < x_1 + b_x \\ y_1 - a_y < y < y_1 + b_y \end{matrix} \right\} \, \Leftrightarrow \, F(x_1,\,y_1) + C_1(a_x, b_x, a_y, b_y) < F(x,\,y) < F(x_1,\,y_1) + C_2(a_x, b_x, a_y, b_y) $$ In other words: $(x, y)$ in some neighbourhood rectangular of $(x_1, y_1)$ if and only if $F(x, y)$ in some neighbourhood (defined with $C_1$ and $C_2$) of $F(x_1, y_1)$.
I have tried some rectangular metrics on $\mathbb{R^2}$ to help, but came up with nothing.