# Can a function be of class C1 even if its partial derivatives are not continuous?

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!

• 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. – Ted Shifrin Apr 28 '19 at 21:29

No. A function of class $$C^{1}$$ iff it has continuous partial derivatives. You can refer to Rudin's book for a proof.