Math205c Spring Term 2003

Final Examination

1. Let (M, d) be a metric space. Show that the following are equivalent: (i) M is connected. (ii) There does not exist a continuous surjection f : M → {0, 1}.

2. Let T_{i}, i = 1, 2, be contractions on a metric space (M, d) with constant c < 1. Assume that

d(T_{1}(x), T_{2}(x)) ≤ ε , x ∈ M ,

and show that d(x_{1}, x_{2}) ≤ \frac{ε}{1−c} for the fixpoints x_{1} and x_{2}.

3. Does there exist a continuous function f : M → N between compact metric spaces which doesn’t map Cauchy sequences to Cauchy sequences? If yes, give an example. If no, justify your answer.

4. Classify the extrema of the function

f : ℝ^{2} → ℝ , (x, y) ↦ \frac{y}{1 + x^{2} + y^{2}} .

5. Determine whether

f : ℝ^{2} → ℝ^{2} , (x, y) ↦ (2xe^{y}, ye^{x}) = (u, v)

can be locallly inverted about the origin and compute ∂_{v}x(0, 0).

6. A surface S is given by

(x, y, z) = (u + sin(v), v + cos(u), e^{uv}) , (u, v) ∈ ℝ^{2} .

Is it the graph of a single function f : ℝ^{2} → ℝ?