Help Understanding Gradients

Question

I understand that gradients are vectors with partial derivatives as components when working in 3D space, but does the the existence of a gradient at a point imply continuity at that point?

Answer 1

No, see this question, it gives a counter-example of a function with partial derivatives at the origin that is not continuous. This is because not every path to the origin yields the same limit.

Answer 2

Wikipedia's Differentiable function article has this neat 2-d example:

$f(x,y) = \begin{cases}x & \text{if }y \ne x^2 \\ 0 & \text{if }y = x^2\end{cases}$

The gradient is uniquely defined at point $(0, 0)$, unaffected by the choice of axes along which you take the partial derivatives, but the discontinuity also exists to the limiting point $(0, 0)$. @TonyK comments that this function is continuous at $(0, 0)$. How about then:

$f(x,y) = \begin{cases}1 & \text{if }y \ne x^2 \text{ or } x = 0 \\ 0 & \text{if }y = x^2 \text{ and } x \ne 0\end{cases}$