In this post I want to discuss some possible approaches to teaching function limits and continuity in terms of epsilon and delta.
The “fear of epsilon and delta” that I refer to here is not only that of the student, but also that of the teacher! Because of the difficulties students have with understanding and working with the multi-quantifier definitions of function limits, continuity and uniform continuity in terms of epsilon and delta, it is very tempting to find a way round the issue. For example, once the students have been taught about convergence of sequences, we can use sequence-based definitions of function limits and continuity. (See my earlier post, however, on teaching the material on convergence of sequences. )
In my second-year mathematical analysis course I have emphasised the sequence approach for many years, while still exposing the students to the epsilon-delta definitions. However, I am beginning to change my views. I no longer see it as desirable to find ways round these problems. I prefer to seek ways to help the students to gain a solid understanding of the harder concepts.
To begin with, let us restrict ourselves to the case of a function from to .
Obviously this should be modified in appropriate ways when dealing with function limits (where the function need not be defined at the point in question) or dealing with functions between more general sets (e.g. subsets of , or more general metric spaces).
In this setting, in terms of sequences, the definition of the statement is continuous is certainly very clean. Here is one version:
For every convergent sequence , we have
This is short and elegant, and it only requires an understanding (see above) of convergence of sequences. This definition is very easy to use to prove results about sums/products/composition of continuous functions (etc.), quoting earlier standard results about the algebra of limits for sequences of real numbers to help where necessary. This then leads to a very clear and clean theory. So why do I have reservations about its use? I have mentioned some of the issues already, but let me list them again here, with some other issues.
- In postponing our confrontation with epsilon and delta for as long as possible, we may not be acting in the best interests of our students.
- Our confidence that students are happy and confident with the notion of convergence of sequences may be misplaced.
- While this definition in terms of sequences is an easy one to use in developing the general theory, on its own it does not give a particularly good introductory, intuitive notion of continuity. Many students have difficult in seeing what this definition really means (in terms of function values being close to the right value) for specific functions such as , and have no idea how one could go about checking carefully that the condition is satisfied. (Obviously you can explain how to do this using, for example, the algebra of limits for this particular function.)
- Students find it difficult to negate this definition correctly when trying to understand what it means for a function to be discontinuous. This is, of course, a general problem. Here it takes the form that students assume that if a sequence does not converge to a particular value, it is because it converges to some other value. The possibility that it does not converge at all may be overlooked.
The function might be a more convincing example than here, except that approaches to teaching the properties of can sometimes be a little circular. Mostly we use the “standard fact” that the derivative of is and the Mean Value Theorem to show that , etc., but obviously we have assumed far more than the continuity of in the process. I am as guilty as anyone else of referring students to books for more details concerning the trigonometrical functions (etc.).
So, what is the problem with the epsilon-delta definition of continuity? Well, if you jump straight in, you end up with a four-quantifier statement. In terms of and delta, “ is continuous” comes out as some variant of the following:
such that we have
Now it is obvious that we could first attack the three-quantifier statement
“ is continuous at “,
and then define continuity in terms of this. Still, a three-quantifier statement is already rather challenging for, say, a first-year student of analysis.
We can also, if we wish, disguise one of the quantifiers by replacing
with the somewhat less formal version
(without openly specifying that is in ).
This is perfectly fine once students have a good understanding of the concepts and can use quantifiers correctly, but I am not convinced that it is wise when students are still in the process of learning how to handle quantifiers.
Rather than disguising this quantifier, I prefer to package it using images of sets, and to say
We can also define continuity of in terms of function limits (one-sided or two-sided). I have no objection to this, but of course it simply shifts the original problem to the alternative setting of function limits.
So, having pointed out problems with these approaches, what is my recommendation? Well, I have some possible ideas involving quantifier packaging, but I haven’t yet had the opportunity to try them out. Perhaps some discussion here would be wise before unleashing them on the students!
I think that whatever approach we take should be applicable to function limits, so let us change our setting slightly.
Let , let be a function from to , and let .
In terms of sequences, the definition of “” is the following:
For every sequence such that converges to , we have as
As before, this is clean, clear, easy to use, and allows you to sidestep the epsilon-delta definition if you wish. The epsilon-delta definition comes out as some variant of the following:
such that we have
or, using images of sets, we have the following
Perhaps some suitable notation for a punctured -neighbourhood would make this look a little less unwieldy.
These are standard, and look easy enough to the professional mathematician, but are still complicated enough to cause serious difficulties for a student trying to come to grips with analysis.
Can we package the quantifiers further? I think that we can, but only at the expense of inventing new terminology. Here is a possible attempt, based on my method for teaching convergence of sequences. I am not at all sure that this is the definitive version. Suggestions are welcome!
Let and let . Then let us say that the set absorbs the values of -near if
We then say that the set absorbs the values of near if there exists a such that absorbs the values of –near .
With this terminology, the definition of “” becomes:
Every open interval centred on absorbs the values of near .
So, after all this work, we are back to a single-quantifier definition. Why not stick to the definition using sequences? Well, we are now in a position to give the students a thorough introduction to the use of epsilon and delta. Time permitting, we can look at lots of examples of sets which do or don’t absorb the values of functions near or -near various points. In the process we can reinforce intuitive notions of function limit and continuity, without sacrificing rigour.
Note that, if we start with , the definition of “ is continuous at ” can now be stated either as the standard
or, with the new terminology, as follows:
Every open interval centred on absorbs the values of near .
Does anyone have any alternative suggestions for names for some of these concepts? Or do they already have names in the literature that I am simply unaware of?
Notes added 5/1/09:
Maybe there is no need to bring in the word “absorbs” here, though it is tempting to make some use of it. If we do use the notion of absorption, it should perhaps be more closely associated with some notion of movement in the domain. For example, we could say that the set absorbs the values as . Alternatively, we could simply say that the set includes the values of for near .
There is a second reason that I am not yet happy with the version above,
absorbs the values of near .
This version appears to be a little ambiguous, and potentially confusing: the values in question could be taken to be
“those values of which are near “.
Whether we use “absorbs”, “includes” or even that dangerous word “contains”, it may be safer to say
“… the values of for near ”
“… the values of near “.
Maybe we could get away with
“… the values of at points near “?
How about the following statement, formalized in terms of as above?
The set includes (all of) the values of at points near .
- Do we need “all of” to make this as clear as possible?
- Is it a good idea to introduce a term such as absorption to make this statement less unwieldy?
- Should we try to introduce a notion of movement as in “as approaches “?
Or maybe it would be best to eliminate the “values” and “points” altogether and go with statements such as
and the version with ,
absorbs -near .
What do people think?