Topology: An Inquiry-Based Approach

We have likely had previous experiences with continuous functions. Continuity is an important consideration in optimization problems because a continuous function attains a maximum value and a minimum value on any closed and bounded interval. Continuous functions also satisfy the Intermediate Value Theorem, that a continuous function \(f\) takes on all values between \(f(a)\) and \(f(b)\) on an interval \([a,b]\text{.}\) An important consequence of the Intermediate Value Theorem is that if \(f\) is a continuous function on an interval and \(f(a)\) and \(f(b)\) have opposite signs, then \(f\) must have a root in the interval \([a,b]\text{.}\) In this section we will begin to explore continuity of functions between metric spaces. Our ultimate goal in future sections is to understand continuous functions well enough that we can define continuity just in terms of open sets.

This involved providing some explanation about what it means for a function \(f\) to have a limit at a point. Intuitively, the idea is that a function \(f\) has a limit \(L\) at \(x=a\) if we can make all of the value of \(f(x)\) as close to \(L\) as we want by choosing \(x\) as close to (but not equal to) \(a\) as we need. To extend this informal notion of limit to continuity at a point we would say that a function \(f\) is continuous at a point \(a\) if if we can make all of the value of \(f(x)\) as close to \(f(a)\) as we want by choosing \(x\) as close to \(a\) as we need (now \(x\) can equal \(a\)).

In order to define continuity in a more general context (in topological spaces) we will need to have a rigorous definition of continuity to work with. We will begin by discussing continuous functions from \(\R\) to \(\R\text{,}\) and build from that to continuous functions in metric spaces. These ideas will allow us to ultimately define continuous functions in topological spaces.

We can picture this as shown at left in Figure 6.1. Here we want to make the values of \(f(x)\) lie within an \(\epsilon\) band of \(f(a)\) above and below \(f(a)\text{.}\) That is, we want to be able to make the values of \(f(x)\) lie between \(f(a)-\epsilon\) and \(f(a)+\epsilon\text{.}\)

Now we have to address the question of how we “make” the values of \(f(x)\) to be within \(\epsilon\) of \(f(a)\text{.}\) Since the values \(f(x)\) are the dependent values, dependent on \(x\text{,}\) we “make” the values of \(f(x)\) have the property we want by choosing the inputs \(x\) appropriately. In order for \(f\) to be continuous at \(x=a\text{,}\) we must be able to find \(x\) values close enough to \(a\) to force \(| f(x) - f(a) | \lt \epsilon\text{.}\) Pictorially, we can see how this might happen in the image at right in Figure 6.1. We need to be able to find an interval around \(x=a\) so that the graph of \(f(x)\) lies in the \(\epsilon\) band around \(f(a)\) for values of \(x\) in that interval. In other words, we need to be able to find some positive number \(\delta\) so that if \(x\) is in the interval \((a-\delta, a+\delta)\text{,}\) then the graph of \(f(x)\) lies in the \(\epsilon\) band around \(y=f(a)\text{.}\) More formally, if we are given any positive tolerance \(\epsilon\text{,}\) we must be able to find a positive number \(\delta\) so that if \(| x-a | \lt \delta\) (that is, \(x\) is in the interval \((a-\delta, a+\delta)\)), then \(| f(x) - f(a) | \lt \epsilon\) (or \(f(x)\) lies in the \(\epsilon\) band around \(y=f(a)\)).

This gives us a rigorous definition of what it means for a function \(f: \R \to \R\) to be continuous at a point.

Note that the value of \(\delta\) can depend on the value of \(a\) and on \(\epsilon\text{,}\) but not on values of \(x\text{.}\)