18. Continuity

18.1 Definition:

Fix a Metric Space $\QTR{Large}{(M,d)}$ and a point

Given 0 we define the open ball of radius around $\QTR{Large}{x,}$
We say is open if for any there exist an 0 such that
The set of open sets is called the Topology defined by the Metric.

18.2 Theorem:

In a Metric Space $\QTR{Large}{(M,d)}$

$\QTR{Large}{M}$ and (the empty set) is open
Let be an arbitrary set of open sets, then is also open.
Let be a finite set of open sets, then is also open.

Proof:

1. and 2. follow directly from the definitions, however if you haven't been through this material you should write down the details.

3. makes use of the observation that if then . Choose . Let

. We have for all $\QTR{Large}{i}$ hence .

18.2.1 Exercise: Show that is open for any 0.

Solution:

For every . We need to find some 0 such that Let and For every . We have

Thus since this is true for any $\QTR{Large}{z.}$ we have

18.3 Definition:

Given Metric Spaces , and a function . We say that $\QTR{Large}{f}$ is continuous a point $\QTR{Large}{x\in }$ if given any 0 there is a 0 such that

We say that $\QTR{Large}{f}$ is continuous if it is continuous at every point.

Exercise: Convince yourself that for this is just the usual definition.

18.3.1 Theorem:

Given Metric Spaces , ,a function is continuous if and only for every open set

is open in

Proof:

Assignment: To be turned in one week from today.

Solution:

Suppose for every open set we know that is open in Suppose we are given 0 and $\QTR{Large}{x\in }$ . We need to find 0 such that . But by 18.2.1 we know that is open. Hence is open. Hence we can find 0 such that . Or, equivalently,

Suppose is continuous and is open we need to show that is open in Suppose we are given 0 and $\QTR{Large}{x\in }$ such that Since $\QTR{Large}{f}$ is continuous, we can find 0 such that $\$

Thus

18.4 Corollary

Suppose we have two metrics and defined on that produce the same Topology (open sets). Then a function is continuous with respect to the metric if and only if it is continuous with respect to the metric

Exercise: State and prove the corresponding appropriate result for

18.5 Corollary

Suppose we have three Metric Spaces , , and suppose and are continuous, then so is

Proof:

Let be open. Since $\QTR{Large}{g}$ is continuous so is is open in and since $\QTR{Large}{f}$ is continuous

so is

18.6 Notation:

Given Metric Spaces , we use the notation , to represent the set of continuous functions from to

18.7 Lemma:

Let be a metric space and be a continuous function from to , then
, ,
where, as in 0.1,
Let be a metric space and be a continuous function from to , then
, ,
where

Proof: Immediate from 18.5

Returning to 17.3 The Circle, we want to show that the three Metrics

produce the same set of continuous functions.

Assignment: (To be turned in one week from today) We could simply use 18.4 (do this!)

Solution:

We need to show that is open with respect to iff it is open with respect to iff it is open with respect to

To accomplish this, it suffices to show that if we are given $\QTR{Large}{x\in }$ and 0 we can find, in sequence 0 , 0 and 0 such that

One note: there is no loss of generality in assuming that the or we work with for the next computation are smaller than the one we initially are given, or compute.

Beginning with a simple case, let Suppose we know that then we know

So .

The hardest calculations involve . Given 0 , I need find 0

$\vspace{1pt}$

In computing we make use of the inequality for small positive values of .

Let

then $\QTR{Large}{g(}$ 0 $\QTR{Large}{)=\ }$ 0 and 0 for small

Now let . Suppose

we have or

The computation of follows in a similar fashion using the observation that for 0.

Let and assume that , we have

but it will be useful in the sequel if we go about it in a slightly different way.

18.8 Lemma:

The identity map is continuous with respect to any pair of the above Metrics.

Proof:

Using to denote the identity map when considered as a map between specific Metric spaces, it is sufficient to show

are continuous since for example, is continuous by 18.5. Since all three arguments are similar we give the details of 1. Choose 0 and . Let . Suppose . we have For large radii points can be far apart in distance yet close in terms of angle.

18.9 Theorem:

Let be a map of sets. suppose is equipped with a Metric $\QTR{Large}{d_{1}}$ . Then if $\QTR{Large}{f}$ is continuous with respect to any of the three metrics is continuous with respect to the other two.

Proof:

Suppose ,

As a set map on the other hand, by 18.7 is continuous for