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If partial derivatives exist and continuous in some open set, then function is differentiable at any point of this set

But for function of 1 variable we have:

If derivative exist in some open set, then function is differentiable at any point of this set

The question is, why there is no need in continuity of derivative for 1 variable function?

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The first result you cite for multi-variable functions is a theorem. The second condition you cite for single variable functions is a definition.

Having discontinuous partials does not preclude a function from being differentiable in the multi-variable case (for example, see here), i.e. the condition is sufficient but not necessary. There is nothing about the continuity of the partials (although differentiability does imply the continuity of the function itself) in the definition of differentiability.

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  • $\begingroup$ Thank you for clarification $\endgroup$
    – eps_del
    Apr 21, 2013 at 6:34

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