Partitions of finite nonnegative integer sets with identical representation functions

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let the representation function $R_S(n)$ denote the number of solutions of the equation $n=s+s^{\mathrm{\prime }}$ with $s,s^{\mathrm{\prime }}\in S$ and $s<s^{\mathrm{\prime }}$. In this paper, we determine the structure of $C,D\subseteq \mathbb{N}$ with $C\cup D=[0,m]$ and $\mathrm{\mid }C\cap D\mathrm{\mid }=2$ such that $R_C(n)=R_D(n)$ for any nonnegative integer $n$.