Metric Space: Continuity
Definition
Let's recall the idea of continuity of functions. Continuity means, intuitively, that you can draw a function on a paper, without lifting your pen from it. Continuity is important in topology. But let's start in the beginning:
The classic delta-epsilon definition: Let
be spaces. A function
is continuous at a point
if for all
there exists a
such that: for all
such that
, we have that
.
Let's rephrase the definition to use balls: A function
is continuous at a point
if for all
there exists
such that the following holds: for every
such that
we have that
. Or more simply: 
Looks better already! But we can do more.
Definitions:
- A function is continuous in a set S if it is continuous at every point in S.
- A function is continuous if it is continuous in its entire domain.
Proposition: A function
is continuous, by the definition above
for every open set
in
, The inverse image of
,
, is open in
.
Note that
does not have to be surjective or bijective for
to be well defined. The notation
simply means
.
Note that
Proof: First, let's assume that a function
is continuous by definition (The
direction). We need to show that for every open set
,
is open.
Let
be an open set. Let
.
is in
and because
is open, we can find and
, such that
. Because f is continuous, for that
, we can find a
such that
. that means that
, and therefore,
is an internal point. This is true for every
- meaning that all the points in
are internal points, and by definition,
is open.
(
)On the other hand, let's assume that for a function
for every open set
,
is open in
. We need to show that
is continuous.
For every
and for every
, The set
is open in
. Therefore the set
is open in
. Note that
. Because
is open, that means that we can find a
such that
, and we have that
.