## Devlin's Angle |

In everyday speech, a 'continuous' process is one that proceeds without gaps of interruptions or sudden changes. Roughly speaking, a functionAs 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:y = f(x)is continuous if it displays similar behavior, that is, if a small change inxproduces a small change in the corresponding valuef(x).

A function *f* is continuous at a number *a* if the
following three conditions are satisfied:

1.To make this precise, we need to define the notion of a limit:fis defined on an open interval containinga.2.

f(x)tends to a limit asxtends toa.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 < |With 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 notx-a| < \delta, then |f(x)-L| < \epsilon

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 values2. 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.

This article is based on the paper

Devlin's Angle is updated at the beginning of each month.