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FE R^2->R, bounding absolute value

Source: VTRMC 2015 P6

May 5, 2021
functional equationfeabsolute value

Problem Statement

Let (a1,b1),…,(an,bn)(a_1,b_1),\ldots,(a_n,b_n) be nn points in R2\mathbb R^2 (where R\mathbb R denotes the real numbers), and let ϵ>0\epsilon>0 be a positive number. Can we find a real-valued function f(x,y)f(x,y) that satisfies the following three conditions?
1. f(0,0)=1f(0,0)=1; 2. f(x,y)≠0f(x,y)\ne0 for only finitely many (x,y)∈R2(x,y)\in\mathbb R^2; 3. ∑r=1n∣f(x+ar,y+br)−f(x,y)∣<ϵ\sum_{r=1}^n\left|f(x+a_r,y+b_r)-f(x,y)\right|<\epsilon for every (x,y)∈R2(x,y)\in\mathbb R^2.
Justify your answer.
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