| | Devlin's Angle |
What exactly is a continuous function? Here is a typical explanation taken from a university textbook (G. F. Simmons, Calculus with Analytic Geometry, McGraw-Hill, 1985):
In everyday speech, a 'continuous' process is one that proceeds without gaps of interruptions or sudden changes. Roughly speaking, a function y = f(x) is continuous if it displays similar behavior, that is, if a small change in x produces a small change in the corresponding value f(x).As the author observes, this description is "rather loose and intuitive, and intended more to explain than to define." He goes on to provide a "more rigorous, formal definition," which I summarize as:
A function f is continuous at a number a if the following three conditions are satisfied:
1. f is defined on an open interval containing a.To make this precise, we need to define the notion of a limit:2. f(x) tends to a limit as x tends to a.
3. That limit is equal to f(a).
If a function f(x) is defined on an open interval containing a, except possibly at a itself, we say that f tends to a limit L as x tends to a, where L is a real number, if, for any \epsilon > 0, there is a \delta > 0 such that:
if 0 < |x - a| < \delta, then |f(x) - L| < \epsilonWith limits defined in this way, the resulting definition of a continuous function is known as the Cauchy-Weierstrass definition, after the two nineteenth century mathematicians who developed it. The definition forms the bedrock of modern real analysis and any standard "rigorous" treatment of calculus. As a result, it is the gateway through which all students must pass in order to enter those domains. But how many of us manage to pass through that gateway without considerable effort? Certainly, I did not, and neither has any of my students in twenty-five years of university mathematics teaching. Why is there so much difficulty in understanding this definition? Admittedly the logical structure of the definition is somewhat intricate. But it's not that complicated. Most of us can handle a complicated definition provided we understand what that definition is trying to say. Thus, it seems likely that something else is going on to cause so much difficulty, something to do with what the definition means. But what, exactly?
Let's start with the intuitive idea of continuity that we started out with, the idea of a function that has no gaps, interruptions, or sudden changes. This is essentially the conception Newton and Leibniz worked with. So too did Euler, who wrote of "a curve described by freely leading the hand." Notice that this conception of continuity is fundamentally dynamic. Either we think of the curve as being drawn in a continuous (sic) fashion, or else we view the curve as already drawn and imagine what it is like to travel along it. This means that our mental conception has the following features:
1. The continuous function is formed by motion, which takes place over time.Aspects of this dynamic view are still present when we start to develop a more formal definition: we speak about the values f(x) approaching the value f(a) as x approaches a. The mental picture here is one of preserving closeness near a point.2. The function has directionality.
3. The continuity arises from the motion.
4. The motion results in a static line with no gaps or jumps.
5. The static line has no directionality.
Notice that the formal definition of a limit implicitly assumes that the real line is continuous (i.e., gapless, or a continuum). For, if it were not, then talk about x approaching a would not capture the conception we need. In this conception, a line or a continuum is a fundamental object in its own right. Points are simply locations on lines.
When we formulate the final Cauchy-Weierstrass definition, however, by making precise the notion of a limit, we abandon the dynamic view, based on the idea of a gapless real continuum, and replace it by an entirely static conception that speaks about the existence of real numbers having certain properties. The conception of a line that underlies this definition is that a line is a set of points. The points are now the fundamental objects, not the line. This, of course, is a highly abstract conception of a line that was only introduced in the late nineteenth century, and then only in response to difficulties encountered dealing with some pathological examples of functions.
When you think about it, that's quite a major shift in conceptual model, from the highly natural and intuitive idea of motion (in time) along a continuum to a contrived statement about the existence of numbers, based on the highly artificial view of a line as being a set of points. When we (i.e., mathematics instructors) introduce our students to the "formal" definition of continuity, we are not, as we claim, making a loose, intuitive notion more formal and rigorous. Rather, we are changing the conception of continuity in almost every respect. No wonder our students don't see how the formal definition captures their intuitions. It doesn't. It attempts to replace their intuitive picture with something quite different.
Perhaps our students would have less trouble trying to understand the Cauchy-Weierstrass definition if we told them in advance that it was not a formalization of their intuitive conception -- that the mathematician's formal notion of a continuous function is in fact something quite different from the intuitive picture. Indeed, that might help. But if we are getting into the business of open disclosure, we had better go the whole way and point out that the new definition does not explicitly capture continuity at all. That famous -- indeed, infamous -- epsilon-delta statement that causes everyone so much trouble does not eliminate (all) the vagueness inherent in the intuitive notion of continuity. Indeed, it doesn't address continuity at all. Rather, it simply formalizes the notion of "correspondingly" in the relation "correspondingly close." In fact, the Cauchy-Weierstrass definition only manages to provide a definition of continuity of a function by assuming continuity of the real line!
It is perhaps worth mentioning, if only because some students may have come to terms with the idea that a line is a set of points, that in terms of that conception of a line -- which is not something that someone or something can move along -- the original, intuitive idea of continuity reduces simply to gaplessness. In short, however you approach it, the step from the intuitive notion of continuity to the formal, Cauchy-Weierstrass definition, involves a huge mental discontinuity.