The following graphs illustrate examples from the additional notes on differentiability, Section 14.4.
Example 2.2
$f(x,y) = \sqrt{x^2+y^2}$ is not differentiable at $(0,0)$.
Example 2.4
$f(x,y) = \frac{xy}{x^2+y^2}$ is not differentiable at $(0,0)$.
Example 2.5
$f(x,y) = \frac{y^3}{x^2+y^2}$ is not differentiable at $(0,0)$.