For a nonempty set A of integers and any integer n, denote r_A (n) by the number of representations of n of the form n=a+a', where a≤a' and a, a'∈A and d_A (n) by the number of pairs (a, a') with a, a'∈A such that n=a-a'. In 2008, Nathanson considered the representation function with infinitely many zeros. As a main result, we prove that, for any set T of integers with density zero, there exists a sequence A of integers such that r_A (n)=1 for all integers n∉T and r_A (n)=0 for all integers n∈T, and d_A (n)=1 for all positive integers n. In this talk, we will present our recent results on representation functions.