Abstract
D. Preiss proved that the graph of the derivative of a continuous Gâteaux-differentiable function f : ℝ2 → ℝ is always connected. We show that this is no longer true in higher dimensions: we construct a continuous, Gâteaux-differentiable function f : ℝ3 → ℝ for which the range of its gradient mapping {∇ f(x) : x ∈ ℝ3} is disconnected. We also give an example of an approximately differentiable continuous function on ℝ2 such that the range of its gradient mapping is disconnected.
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