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I know that if all partial derivatives of a function f exist and are continuous then the function is said to be of class C1 (continuously differentiable).

However, I was not able to find whether this is a necessary or a sufficient condition. What I mean is: Could a function be of class C1 even though its partial derivatives are not continuous?


PS. Can a function be continuous/differentiable even if its partial derivatives are not continuous?

Thank you!

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  • $\begingroup$ If a function $f\colon \Bbb R^n\to\Bbb R^m$ has a derivative, that derivative is represented in matrix form (with respect to the standard bases) by the partial derivatives. $\endgroup$ – Ted Shifrin Apr 28 '19 at 21:29
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No. A function of class $C^{1}$ iff it has continuous partial derivatives. You can refer to Rudin's book for a proof.

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  • $\begingroup$ What needs proof is the converse! $\endgroup$ – Ted Shifrin Apr 28 '19 at 21:28

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