Metric Space: Continuity

                     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 .

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 .