Math 410 Midterm Exam
Instructions: Due Thursday March 3. You may use your class
notes and the textbook, but no other resources are allowed. Do not
discuss the problems with anyone else.
(1) Let X be a compact metric space. Let U be an open covering of
X. Prove that there exists a δ > 0 with the following property:
for every x ∈ X there exists some U ∈ U such that the δneighborhood about x is contained in U .
(2) Let X be a metric space. Let A and B be compact subsets of X
which are disjoint. Prove that there exist disjoint open subsets
U and V of X with A ⊆ U and B ⊆ V .
(3) Let X be a metric space and let A be a subset of X. Let
C be a connected subspace of X such that C ∩ A 6= ∅ and
C ∩ (X − A) 6= ∅. Prove that C ∩ Bd(A) 6= ∅ (here Bd(A)
denotes the boundary of A).
(4) Let X be a metric space. Let A0 , A1 , A2 , . . . be a sequence of
connected subspaces
S of X. Assume that An ∩ An+1 6= ∅ for all
n ≥ 0. Prove that n An is connected.
(5) Let x0 ∈ Rn and let > 0. (a) Prove that the -cube C (x0 ) =
{x ∈ R2 : d∞ (x, x0 ) < } about x0 is convex.
(6) Let f : R2 → R be given by f (x, y) = |x| + |y|. (a) For which
vectors u 6= 0 does f 0 (0; u) exist? Evaluate it when it exists.
(b) Is f differentiable at 0? (c) Is f continuous at 0? Prove all
of your assertions.
(7) Let A ⊆ Rm be open and let f : A → R. Suppose that the
partial derivatives Dj f exist and are bounded in a neighborhood
of a. Prove that f is continuous at a.
(8) Define f : Rn → Rn by f (x) = kxk22 x. (a) Show that f is class
C 1 . (b) Show that f carries the set B = {x ∈ Rn : kxk2 < 1}
bijectively onto itself. (c) Show that f −1 : B → B is not
differentiable at 0.
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