The authors study the range of the derivative of a Gâteaux-differentiable function \(f\) between Banach spaces with a particular emphasis on the phenomena that may happen when \(f\) is Gâteaux-differentiable but fails to be Fréchet-differentiable. The authors prove that there is a Gâteaux-differentiable Lipschitz function from \(l_1\) to \(\mathbb R^2\) whose derivative ``jumps'' everywhere.