Calculus on Manifolds Midterm Exam April 25, 2012 1. (10 points) If A ⊂ [0, 1] is the union of open intervals (ai, bi) such that each rational number in
(0, 1) is contained in some (ai, bi), show that the boundary ∂A = [0, 1] \ A.
2. (a) (10 points) If A ⊂ Rn is closed and x /∈ A, prove that there is a number d > 0 such that
|y − x| ≥ d for all y ∈ A.
(b) (10 points) Let A, B ⊂ Rn. If A is closed, B is compact, and A ∩ B = ∅, prove that there is d > 0 such that |y − x| ≥ d for all y ∈ A and x ∈ B.
(c) (10 points) Let U ⊂ Rn. If U is open and C ⊂ U is compact, show that there is a compact set D such that C ⊂ interior D and D ⊂ U.
3. Let A ⊂ Rn, and f : A → R be a bounded function. For each δ > 0, let M(a, f, δ) = sup{f (x) : x ∈ A and |x − a| < δ}
m(a, f, δ) = inf{f (x) : x ∈ A and |x − a| < δ}
The oscillation o(f, a) of f at a is defined by o(f, a) = lim
δ→0
£M(a, f, δ) − m(a, f, δ)¤ . (a) (10 points) Show that this limit exists.
(b) (10 points) Show that the bounded function f is continuous at a if and only if o(f, a) = 0.
4. (10 points) Define f : R → R by
f (x) = (
e−x−2 if x 6= 0,
0 if x = 0.
Show that f is a C∞ function, and f(i)(0) = 0 for all i.
5. (10 points) Let A ⊂ Rn be a rectangle and let f : A → Rn be continuously differentiable. If there is a number M such that |Djfi(x)| ≤ M for all x in the interior of A, prove that
|f (x) − f (y)| ≤ n2M |x − y|
for all x, y ∈ A.
6. (a) (10 points) State the Inverse Function Theorem.
(b) (10 points) State the Implicit Function Theorem.
