Conditions for differentiability for many variables

For single variable functions we learn that being differentiable implies in being continuous. We also learn that being continuous doesn't imply in being differentiable because there are some exceptions to the rule. For multivariable functions differentiability also requires continuity, but we also have specific cases where we can calculate partial derivatives while the function is also discontinuous at a point.

[math]\displaystyle{ f(x) = |x| }[/math]. This single variable example shows that the sided limits exist, the function is continuous at the origin and yet, we can't differentiate it at the origin because the left and right limits yield different results. Visually there is a sharp edge at the origin. [math]\displaystyle{ f(x,y) = \sqrt{x^2 + y^2} }[/math]. The same behaviour in two variables. The function is continuous at the origin, but the sharp edge means that we can't differentiate it there.

[math]\displaystyle{ f(x,y) = \begin{cases} 0 & \text{if} & (x,y) = (0,0) \\ \frac{xy}{x^2 + y^2} & \text{if} & (x,y) \neq (0,0) \end{cases} }[/math] If we calculate [math]\displaystyle{ \frac{\partial f}{\partial x} }[/math] and [math]\displaystyle{ \frac{\partial f}{\partial y} }[/math] we do find valid results. But we also know, from calculating limits with different paths, that the function is discontinuous at the origin.