Connected Sets

We have a few details to clean up before we move on to other things; these concern the special role of intervals in ℝ and their use in characterizing the open sets in ℝ given in Chapter Four (see Theorem 4.6 and Exercise 4.25). As we'll see in this section, a better understanding of the special nature of intervals in ℝ will allow us to generalize the intermediate value theorem of calculus. The intermediate value theorem is the formal statement of the informal notion that the graph of a continuous function is “unbroken.” The historical basis of the theorem is the concept of a function as measuring, over time, the relative position of an object moving along a straight line. Thus, if we track the position y = f(x) of a moving object between time x = a and some subsequent time x = b, we would expect the object to “visit” all of the positions y that are intermediate to f(a) and f(b). In short, the continuous image of the time interval [a, b] should contain (at least) the full interval of positions between f(a) and f(b).

The secret here is the intuitively obvious fact that no interval in ℝ can be split into two relatively open parts. Let's prove this by “brute force” for the interval [a, b] (we'll do the other cases shortly).