VTRMC 2015 Problem 6

Let $(a_1,b_1),\ldots,(a_n,b_n)$ be $n$ points in $\mathbb R^2$ (where $\mathbb R$ denotes the real numbers), and let $\epsilon>0$ be a positive number. Can we find a real-valued function $f(x,y)$ that satisfies the following three conditions?

1. $f(0,0)=1$ ;
2. $f(x,y)\ne0$ for only finitely many $(x,y)\in\mathbb R^2$ ;
3. $\sum_{r=1}^n\left|f(x+a_r,y+b_r)-f(x,y)\right|<\epsilon$ for every $(x,y)\in\mathbb R^2$ .

Justify your answer.