This applet looks at functions written in polar form.
In particular, it looks at the function f(r,theta)=cos(m*theta)*r^n, where m and n can be modified.
Functions of this form are particularly useful when considering what
can go wrong with continuity and differentiability. In all cases
the key point will be the origin.
Consider when m=3.
If n=0 the function is not continuous.
If n=1/2 the partials don't exist since the slices have cusps.
If n=1, we have tangent lines in all directions, but no tangent plane.
If n=3/2, the function is differentiable.
If n=2, we have the standard monkey saddle.
To convert the functions to the more typical x-y format, recall
that cos(m*theta) is the real part of (cos(theta)+i*sin(theta))^m,
and that r^2=(x^2+y^2).