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.

Any vector-valued function ${\bf g}=(g_1,\ldots,g_m)$ is continuous (at some point $p$ or overall) iff all its component functions $g_i$ are continuous. This is an immediate consequence of the following inequalities for vectors ${\bf x}$, ${\bf y}$ in ${\mathbb R}^n$:
$$|x_i-y_i|\leq|{\bf x}-{\bf y}|\leq\sum_{k=1}^n|x_k-y_k|\ .$$
In particular ${\bf g}:=\nabla f$ is continuous iff all coordinates ${\partial f\over\partial x_k}$ of $\nabla f$ are continuous.
