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I'm a bit confused about this. I know that if all partial derivatives exist it's not necessary for function to be differentiable. Usual examples for non differentiable function for which all partial derivatives exist are ones when function is not continuous. So I'm wondering if that implies that partial derivatives are not continuous. If not, can you give me example? I hope you understand my question

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If the partial derivatives were continuous, then the function would be differentiable and thus, continuous. So $$f \text{ has continuous partial derivatives} \Rightarrow f \text{ is differentiable} \Rightarrow f \text{ is continuous} $$

Using the contrapositive:

$$f \text{ is not continuous } \Rightarrow f \text{ is not differentiable} \Rightarrow f\text{ hasn't continuous partial derivatives}$$

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