This can be visualized by the polar form f ( r cos ⁡ ⁡ ( θ θ ) , r sin ⁡ ⁡ ( θ θ ) ) = r 2 sin ⁡ ⁡ ( 4 θ θ ) 4 ; it is everywhere continuous, but its derivatives at (0, 0) cannot be computed algebraically. Rather, the limit of difference quotients shows that f x ( 0 , 0 ) = f y ( 0 , 0 ) = 0 , so the graph z = f ( x , y ) has a horizontal tangent plane at (0, 0), and the partial derivatives f x , f y exist and are everywhere continuous. However, the second partial derivatives are not continuous at (0, 0), and the symmetry fails. In fact, along the x -axis the y -derivative is f y ( x , 0 ) = x , and so: