14.7
Example.
Two points
on a sphere are called
antipodal points if
and
are opposite ends of the same diameter
of the sphere. We will consider the surface of the earth to be a sphere
of radius
. At any fixed time, let
denote the temperature
of the earth at the point
on the surface of the earth. (More precisely,
let
be the number such that the temperature at
is
).
We will show that there are two antipodal points
on the surface of
the earth such that
In fact, we will show that there are two
antipodal points on the equator with the same temperature. We first introduce
a coordinate system so that the center of the earth is at the origin,
and the plane of the equator is the

plane, and the point on the
equator passing through the Greenwich meridian is the point
.
Then the points on the equator are the points
Define a function
by
Thus
We suppose that
is a continuous function on
.
If
then
, so
and
are a pair
of antipodal points with the same temperature. Now
so if
then
and
have opposite signs. Hence by the
intermediate value property, there is a number
such that
, i.e.
Then
and
are a pair of
antipodal points with the same temperature.
14.8
Example.
Let
where
and
are real numbers, and
.
Then there exists some number
such that
.
Proof: I will suppose that for all
and
derive a contradiction. Let
Since
for all
,
is continuous on
R. We know that
Hence
for some
. Then
and
have
opposite signs, so by the intermediate value property there
is a number
such that
. This contradicts
our assumption that
for all
.
14.10
Exercise.
A
Three wires
are joined at a common
point
.
Let
be the Yshaped figure formed by the three wires.
Prove that at any time there are two points in
with the same temperature.
14.11
Exercise.
A
Six wires are joined to form the figure
shown in the diagram.
Show that at any time there are three points in
that have the same temperature. To simplify the problem, you may assume
that the temperatures at
,
,
, and
are all distinct.