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!

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