PROBLEM SET III
DUE FRIDAY,  MARCH
Exercise . List all the vector subspaces of Rm that are open subsets. List all the vector subspaces of Rm that are
closed subsets. Justify your answers.
Exercise . Consider the map g : R2 .
R given by the formula
{ x2 y
if (x, y) ̸= (0, 0);
4
2
g(x, y) := x +y
0
if (x, y) = (0, 0).
For any unit vector v ∈ R2 , compute limε→0 g(εv). Show that g is not continuous at the origin. [In the first semester,
we learned that a function R . R was continuous at a point a ∈ R just in case the le-hand limit and the right-hand
limit both exist and are equal to the value of the function. Do things look different here?]
Exercise . Consider the unit m-sphere in Rm+1 :
{
Sm =
}
m
∑
x2i = 1 .
(x0 , x1 , . . . , xm ) ∈ Rm+1 i=0
Write down a homeomorphism Rm
.
S − {(1, 0, . . . , 0)}.
m
Exercise . Suppose γ : R . Rm+1 a differentiable curve such that for any t ∈ R, one has γ(t) · γ ′ (t) = 0. Show
that the image of γ lies on a sphere
{
}
m
∑
Sm (r) := (x0 , x1 , . . . , xm ) ∈ Rm+1 x2i = r2
i=0
for some radius r ≥ 0.
