Let X,Y be Banach spaces, D an open subset of X and F a locally Lipschitzian mapping from D into Y. In this paper it is proven that if there is a dense \(G_{\delta}\)-subset G of D such that there exist all the first partial derivatives of F at x for all \(x\in G\), then F is \(\hat Ga\)teaux differentiable in \(G_{\delta}\). Some similar results were obtained under different conditions by P. Kenderov [Proc. Second Spring Conf. Bulg. Math. Soc., Vidin 1973, 123-126 (1974; Zbl 0261.46005)] and by K.-S. Lau and C. E. Weil [Proc. Amer. Math. soc. 70, 11-17 (1978; Zbl 0391.26004)].