A property of partial derivatives (Q1108550)

Let X, Y be topological vector spaces, \(Z=X\times Y\), V a locally convex space and f:Z\(\to V\) a separately weakly differentiable mapping. If \(D_ 1f(z)=0\) or \(D_ 2f(z)=0\) for each z in Z, then \(D_ 1f(z)=0\) for all z in Z or \(D_ 2f(z)=0\) for all z in Z. This result is no longer valid if V is not locally convex. An analogoue of this result for three or more variables fails.