On the range of the derivative of Gateaux-smooth functions on separable banach spaces

摘要

We prove that there exists a Lipschitz function froml 1 into ℝ2 which is Gâteaux-differentiable at every point and such that for everyx, y εl 1, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gâteaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gâteaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0.