Consider a sequence of real-valued functions \(\{f_k, k\geq 1\}\) defined on a closed subset \(S\) of the real line. This sequence is called convergence-preserving on \(S\) if \(\{f_k(t_k), k\geq 1\}\) is convergent for every convergent sequence \(\{t_k\} \subset S\). The main result shows that \(\{f_k\}\) is convergence-preserving on a closed interval \([a,b]\) if and only if the sequence \(\{f_k\}\) converges uniformly on \([a,b]\) to a continuous function.