Suppose that the partial derivative with respect to the first variable vanishes, i.e., Let be the projection onto the last n -1 components in . Since E is convex, the line joining the two points x and is contained in E . On the other hand, if E is closed and is not connected for some , then we can find a point such that contains a point larger than as well as a point smaller than . Since E is closed, there exists a small interval containing which does not intersect . Setting yields a function which clearly depends on the value of x 1 . 6.
Theorem Let be open and connected.
Assume that that is differentiable, with Then f is constant on U . Observe that for every , the open set U contains a small ball B of positive radius about b . By Theorem 9.19 and its Corollary, f is constant on B . On the other hand, is also open, being the inverse image of an open set under a continuous function. Since U is connected, it cannot be decomposed into two nonempty disjoint open sets.