In this note, we extend the result in Lepetier and San Martin (Stat. Probab. Lett. 32:425-430, 1997) by eliminating the condition that (g(t, 0, 0)) t∈[0,T ] is a bounded process. Furthermore, we prove that if g is Lipschitz continuous in y and uniformly continuous in z, and (g(t, 0, 0)) t∈[0,T ] is square integrable, then for each square integrable terminal condition ξ , there exists a unique square integrable adapted solution to the one-dimensional backward stochastic differential equation (BSDE) with the generator g, which generalizes the corresponding (one-dimensional) results in Pardoux and Peng (Syst.