An example of a function  f  of two variables which is not differentiable at the origin but both the partial derivatives ¶ f/¶ x and ¶ f/¶ y exist (and are zero) at the origin. It is easy to see that  ¶ f/¶ x = 0  and  ¶ f/¶ y = 0  since the function is zero on the x-axis and y-axis away from the origin.