differentiability of partial derivatives Prove that if f a function of n variables is continuously differentiable in an open subset U of $R^n$ then the partial derivatives of f are continuously differentiable. I used the definition of f being differentiable to prove the converse of the statement and that each partial derivative exists but how to prove they are continuously differentiable? 

This is not true.  Let $n = 1$ and let $g:\mathbb R \to \mathbb R$ be continuous but not differentiable.  Let $f(x) = \int_0^x g(s) \, ds$.  Then $f$ is continuously differentiable, but $f'$ is only continuous, not differentiable.
