Partial Derivatives Continuous does not guarantee Gradient Function continuous Just want to check my understanding that partial derivatives continuous does not mean that the gradient function $\nabla f$ is continuous. Is that correct?
E.g. $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ continuous does not necessarily mean that $\nabla f=(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})$ is continuous.
I concluded that based on "Continuity in each argument is not sufficient for multivariate continuity".
Question 2) What conditions guarantee that $\nabla f$ is continuous?
Thanks for any help.

One way things can go wrong is if the partial derivatives exist but the gradient does not. For instance, $$ f(x,y) = \left\{ \begin{array}{cl} \dfrac{xy}{x^2 + y^2} & \text{if }(x,y) \not= (0,0) \\ 0 & \text{if } (x,y) = (0,0) \end{array} \right.$$ 
has the property that $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ exist and are continuous at all points $(x,y)$, but $f$ has no gradient at the origin as it fails even to be continuous there.
It is for this reason that writing $\nabla f = ( \frac{\partial f}{\partial x},\frac{\partial f}{\partial y})$ requires a small amount of caution.
