On the range of the derivative of Gâteaux-smooth functions on separable Banach spaces

We prove that there exists a Lipschitz function from l into IR which is Gâteaux-differentiable at every point and such that for every x, y ∈ l, the norm of f (x) − f (y) is bigger than 1. On the other hand, for every Lipschitz and Gâteaux-differentiable function from an arbitrary Banach space X into IR and for every ε > 0, there always exists two points x, y ∈ X such that ‖f (x)−f (y)‖ is less than ε. We also construct, in every infinite dimensional separable Banach space, a real valued function f on X, which is Gâteaux-differentiable at every point, has bounded non-empty support, and with the properties that f ′ is norm to weak continuous and f (X) has an isolated point a, and that necessarily a 6= 0. 1991 Mathematics Subject Classification : primary : 46 B 20, secondary : .