On the primitives of differential one-maps (Q2754975)

Let \(f\) be a function having its domain and range in \(\mathbb{R}^n\). Then it is well-known that if the mapping \(f\) is continuously differentiable on a star-like open set \(U\) then it has a primitive if and only if its Jacobian is symmetric at each point. A completely analogous theorem is true for the differential one-maps between Banach spaces: if a differential one-map between Banach spaces is continuously differentiable then it has a primitive if and only if its derivatives as a bilinear form are symmetric. In this paper, the author gets its generalization without assuming the continuity of the derivative. That is, making use the idea of the Goursat Lemma, he proves that if the function is continuous on a star-like open set and is differentiable possibly except of one point, then it has a primitive if and only if its derivatives as a bilinear form are symmetric at every point where it exists. So, his proof is not different from the usual ones which depend on the variational equations or the Frobenius-Dieudonné Theorem. A generalization of the theorem on the existence of the potential is also obtained.