## Suppose that F: Rn -> Rn is continuously differentiable everywhere

Definition: Suppose that F: Rn -> Rn is continuously differentiable everywhere. A point P∈R^n is called an isolated singularity of F if DF_p is not invertible but DF_y is invertible for all Y≠P in some neighborhood of P.

a. Let f: R -> R by f(x) = 3 x^4 - 20 x^3. Prove that f has exactly two points which are isolated singularities. Describe what happens at each point.

b. Let f denote the transformation that takes rectangular coordinates into polar coordinates. Does f have any isolated singularities? If so, identify them and explain what happens at each point?

c. Write down explicitly a continuously differentiable function F : R2 -> R2 that has an isolated singularity at the origin and no other singularity.