Due on September 26th
Problem 1 (*): Fix an integer 1 ≤ k ≤ n. Prove that the “projection map”
πk : Rn → R defined by the rule,
πk (x1 , x2 , ..., xn ) = xk
is a continuous map at any point p ∈ Rn by using the δ, ϵ definition.
Problem 2 (*): Let || · || be any norm on Rn . Prove that the norm map,
|| · || : Rn → R is a continuous map at any point p ∈ Rn . Use the δ, ϵ definition.
Problem 3 (*): Let f : Rn → Rm be any map. Prove that f is continuous if
and only if (πk ◦ f ) is continuous for every 1 ≤ k ≤ n.
Problem 4 (*): Suppose f : A → B is a continuous function (between any two
regions which might come from different Euclidean spaces). Suppose (an )n≥0 is a
sequence such that an ∈ A and the sequence converges to a point p ∈ A. Prove
that (f (an ))n≥0 is a sequence which converges to f (p) ∈ B.
Note: This is the old-familiar limx→p f (x) = f (p) that you seen in first-semester
calculus. But the above paragraph is the precise statement of this result.
Problem 5 (**): Recall that a function f : A → B is “uniformly continuous”
iff for any ε > 0 there is δ > 0 such that if ||x − y|| < δ and x, y ∈ A then
||f (x) − f (y)|| < ε.
Note: When we prove a function is continuous we pick p ∈ A and ε > 0, the
choice of δ depends on p and ε, so δ = δ(p, ε). “Uniformly continuous” means
that δ = δ(ε), i.e. the same choice for δ can be made uniformly for all points.
Suppose A is compact and f : A → B is continuous. Prove that f is uniformly
continuous.
Hint: Let p ∈ A, since f is continuous at p, there is a δ = δ(p, ϵ) which satisfies
a certain condition. Now construct a family of open discs D(p, δ(p)). Observe
that they are an open cover for A, but since A is compact we can choose a finite
number of them to cover A. Now let δ be the minimum of the radii of the discs.
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Problem 6 (*): We proved in class that f : A → B is continuous if for any open
set O in B we have that f −1 (O) is an open set in A. Now prove the converse.
Hint: To show that f −1 (O) is open we first pick a point p ∈ f −1 (O), so f (p) ∈ O.
But O is open so there is ε > 0 such that D(f (p), ε) ∩ B is contained in B. Now
use continuity to choose an appropriate δ. Argue that D(p, δ) ∩ A is contained in
f −1 (O) and explain why this concludes the proof.
Problem 7 (*): Prove that f : A → B is continuous if and only if for any closed
set C in B we have that f −1 (C) is a closed set in A.
Problem 8 (**): We say a region A ⊆ Rn is “path connected ” if and only it
for any two points p, q ∈ A we can find a continuous map f : [0, 1] → A such
that f (0) = p and f (1) = q. Prove that if A is path connected then A is connected.
Hint: The proof is almost the same as we did in class when we proved that con-
vex sets are connected. Recall in that proof we defined L to be the line segment
between p and q, and we used the fact that L is connected (since it is an inter-
val). In this proof we rather let L = f ([0, 1]), i.e. the push-forward of [0, 1] into
A, and since [0, 1] is connected it means L is also. Now you can complete the proof.
