Let \(G\) be a graph which is simple and finite. Let \(V(G)\) and \(\delta(G)\) denote the set of vertices and the minimum degree of \(G\), respectively. The authors prove that if \(\delta(G)\geq b|V(G)|/(a+ b)\), \(|V(G)|> (a+ b)(a+ b-3)/a\), and there is a function \(f\) from \(V(G)\) into \(\{a,a+1,\dots, b\}\) such that \(\sum_{x\in V(G)} f(x)\) is even, where \(a\), \(b\) are positive integers with \(a\leq b\), then \(G\) has an \(f\)-factor. A similar result was obtained independently by M. Kano and N. Tokushige [J. Comb. Theory, Ser. B 54, No. 2, 213-221 (1992; Zbl 0772.05080)].