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 \((0, 0)\).
