Pro and Against - Continuity and Limits

Arguments (pro/against) regarding teaching continuity and limits
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Transcript Although I never had any difficulty with the ϵ  - δ  definition, I still found that continuity made much more sense after I encountered the general topological setting. However, I ended up a set- theoretic topologist; after some forty years of teaching mathematics, I’m quite certain that this is a minority e xperience. I’m also quite certain that there is no single right answer to the question of how to teach continuity in a first rigorous approach: the answer depends not only on the individual student, but also on the preferences of the instructor. I do think that it’s worth being aware of the range of possibilities and some of their strengths and weaknesses. I’m familiar with five approaches to teaching continuity in a rigorous fashion.  1.   The traditional ϵ  - δ  approach. This has the overwhelming advantage of universal familiarity and use, and the distinct disadvantage that while the intuition underlying continuity is covariant −  if  x  is near a ,  f  (  x )  is near  f  ( a )   −  the definition is contravariant. The definition is also ∀∃∀ , which is logically rather complex. On the other hand, it uses concrete, quantitative measures of approximation, which for many students is a plus, and its connection with uniform continuity (and other stronger forms of continuity) is straightforward. 2.   Sequences first, with ϵ  -  N  , then  f   is continuous iff it preserves limits of sequences. The main advantages are that a sequence is a particularly simple kind of function whose convergence is easily visualized (‘it eventually gets inside any given ϵ  -nbhd and stays there’), and that the definition of continuity is covariant. It’s also relatively easy eventually to move on to the ϵ  - δ  definition of continuity. The problem with this approach is that the ‘thinness’ of sequences tends to obscure what’s really going on, namely, that everything  near a  is being sent near  f  ( a ) . It also doesn’t generalize well to uniform continuity, to put it mildly. 3.   Open sets. In full generality this is simply too abstract for most students in a first rigorous encounter with continuity. If we work only with functions on the reals and limit ourselves to open intervals , it isn’t really much different from the ϵ  - δ  approach. It may make some of the theory just a hair simpler, but the main advantage is that it makes the generalization to R  n  easier: open boxes are more convenient than open (Euclidean) balls. On the other hand, general open intervals aren’t as concrete a notion of approximation as ϵ  -nbhds. 4.   An abstract nearness relation. This approach was outlined in P. Cameron, J. G. Hocking and S. A. Naimpally,   Nearness −  A Better Approach to Continuity and Limits , The American Mathematical Monthly, Vol. 81, No. 7 (Aug. - Sep., 1974), pp. 739-745]. It has the advantage of a covariant definition of continuity, of treating convergence of sequences as just another instance of continuity, and of having a very intuitive underpinning. It has the disadvantage of being very much outside the mainstream, so that one must eventually derive the more usual characterizations of continuity anyway. One  pays for the more intuitive introduction by having to spend extra time introducing the standard approach, and it’s not clear to me whether there’s a net benefit. One definite strength of the approach, however, is that it yields uniform continuity in a very  straightforward way: one simply replaces the notion of a point being near a set with the notion of a set being near another set, thereby getting a  proximity space.  5.   Infinitesimals. The classic example of this approach is H. Jerome Keisler,   Elementary Calculus: An Infinitesimal Approach ; another example is Keith Stroyan,   Foundations of  Infinitesimal Calculus . Pretty much everything that I said about the nearness approach applies here as well, including the ease of defining uniform continuity. The main differences are that the preliminaries are a bit more complicated, but the payoff is in my view a bit greater: rigorous infinitesimals are, I think, more useful than the axiomatized notion of nearness used in (4). There is a strong pragmatic argument for (1) or (2), especially in a school that has a lot of transfer students. One can make pretty good pædagogical arguments for (4) and (5), but in most situations they well may be overridden by practical concerns. In practice some combination of ideas from (1), (2), and (3) is likely to be as effective as anything.    People who go on to become professional users of advanced mathematics generally have a different standard of understandable than people who don't. Frankly, if I were writing an introductory real analysis book for people who could be counted on to do things like hold multiple equivalent definitions of an object in their heads, to the point that they can compare and contrast them and form a favorite, the book I would produce would be very different from the standard real analysis textbook. But that isn't the world in which those  books are written. I guess what I'm saying is although you might feel it's more understandable to do it that way, I suspect that if the books actually did that, many would be flummoxed by open sets and happy when they got to ϵ   and δ . The only people who would be happy in either world are the ones for whom the choice makes no difference. (This reminds me, a little bit, of  people who have a favorite set-theoretic construction of R   from the ring of natural numbers, with pedagogical justifications for their favorite. For most people, any  construction of R   from anything   is going to be a huge stumbling block--- huge, that is, compared to the size of any pedagogical choice made about how to do it.)    The other obvious choices for a definition of continuity require forming a mental image of very large sets: the collection of all open subsets of R  , or the collection of all convergent sequences with a given limit. Many people struggle with conceptualizing such large collections--- and try to base proofs involving them on misconceptions of what an arbitrary element of such a collection must look like. (You will see this no matter when you talk about open sets.)    At most schools, the introductory real analysis class must also accommodate the needs of future teachers of K-12 mathematics, and people whose future jobs will not involve mathematics at all (but whose majors require one advanced math class). Many of these  people will not be happy with any  precise definitions, because definitions are used for writing proofs, and they don't see any point in writing proofs. The current choice is partly an acknowledgement to this reality, I think, because...    ...the ϵ  - δ  definition of continuity is perhaps the closest of many possibilities to the intuitive conception of continuity given in calculus classes (  f   is continuous at c  if for  however strictly you interpret ≈ , you can always ensure that  f  (  x )≈  f  ( c )  by taking  x  sufficiently close to c ). I do think that, for a population of future proof writers, the open sets definition should definitely get more emphasis than playing with ϵ   and δ . (But you can use nothing but   the ϵ  - δ  definition, and still minimize the amount of playing with ϵ   and δ . It comes down to writing style. I can't defend the writing style of many real analysis textbooks, but I feel these issues usually go far beyond just what choices are made in the definitions.)    How do you define an open subset of R  ? Maybe: a set G  is open if for all  g   in G  there are a  and b  in G  satisfying a <  g  < b  and ( a , b ) ⊆ G . OK. To verify from a given subset of R   is open  from this definition  (and not from nice theorems about the sets that satisfy the conditions of this definition), you need to fix  g  's, and produce a 's and b 's making the above claim true. But this is a Roman letter version of what you probably don't like about ϵ   and δ .    Let me expand on my comment.    Towards your question: I think that tolerances ϵ  >0  and allowances δ >0  are things with a size and thus are much more tangible than open sets and other ghosts from general topology.    In my view the basic notion is that of continuity . A function  f   is continuous at  x 0  (think of  x 0 := π  ) if inputting a value  x  near  x 0  results in a function value that is not far off the actual value  f  (  x 0 ) . Now of course we need a numerical version of this idea. One would be happy when    |  f  (  x )−  f  (  x 0 )|≤|  x −  x 0 | ,      i.e., if the error in the output were at most as large as the error in the input, and we would  be content, if there were a constant C  >0  such that    |  f  (  x )−  f  (  x 0 )|≤ C  |  x −  x 0 | .      When  f   satisfies such a condition it is called  Lipschitz-continuous . Unfortunately there are cases where we have continuity in an intuitive sense, but there is no such C  , e.g.,  f  (  x ):=  x √  at 0 . This brings us to a more involved definition … , and on, and on.    Central to all computing is the fact that the basic arithmetic operations in R   and C  are continuous. This is proven via simple inequalities and has as a consequence all the rules about limits of sums etc. we learn later.    Concerning limits: A function  f   has limit η  for  x → ξ   if defining  f  ( ξ  ):= η  would make it continuous there.    One reason for the traditional setup (starting with ε  and δ ) may be that not all questions in real analysis can be reduced to topology alone. Depending on the curriculum, the need to connect the material with prior exposure to calculus may also play a role.    Although the definition of continuity via open sets is elegant and effective, I do not think it to be particularly intuitive for beginners. It also does not give a good description of continuity at a point. Instead I prefer to define continuity via neighbourhoods. This approach is used e.g. in the books of Jameson Topology and normed spaces or Brown Elements of modern Topology . The advantage is that you can start with the notion of a  system of neighbourhoods of a point (the only condition being that a neighbourhood U   of a point  x  must satisfy  x ∈ U  ) and state the definitions for a map to be continuous     f  :  X  → Y   is continuous at a point  x ∈  X   if for every neighbourhood U  ∋  f  (  x )  there is a neighbourhood V  ∋  x  with  f  ( V  ) ⊆ U  .     f  :  X  → Y   is continuous if it is continuous at every point  x ∈  X  .     before you go into the precise conditions for a neighbourhood system. In particular you can take open intervals or open disks in the plane as examples of neighbourhoods and  provide pictures. This can then also be translated into the ε − δ  statements, but the intuition can be built first. shareimprove this answer   
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