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

We prove that there exists a Lipschitz function from ξ ¹ into ℝ ² which is Gâteaux-differentiable at every point and such that for every x, y ∈ l ¹, 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 ℝ and for every ε> 0, there always exist 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 ≠ 0.