Differentiabilty of Complex Function

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My professor was discussing this question. After finding out that CR equations are satisified at origin for the function he used the definition of derivative to check for differentiabilty. Also, he mentioned that we can't compute partial derivatives of u(x,y) and v(x,y), and for example one of them turns out to be discontinious at origin then say that the function is bot differentiable. I couldn't understand his reasoning. Isn't that the sufficient condition for differentiabilty?Thoughts?

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