A function without a total differential

Task number: 3200

The function \(f\colon \mathbb R^2 \to \mathbb R\) is defined as \[ f(x, y) = \sqrt{|x||y|}. \]

Note that \(f\) is continuous at the point \((0, 0)\). At this point calculate \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\). Prove that \(f\) does not have a total differential at the point \((0, 0)\).

    Difficulty level: Easy task (using definitions and simple reasoning)
    Solution require uncommon idea
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