There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The mathematical community often holds an upbeat opinion on the success of student reception of this (see examples below), whereas the education community often stresses difficulties and their "baffling" and "inhibitive" effect (see below). A typical educational perspective on this was recently expressed by Paul Dawkins in the following terms:

2.3. Student difficulties with real analysis definitions. The concepts of limit and continuity have posed well-documented difficulties for students both at the calculus and analysis level of instructions (e.g. Cornu, 1991; Cottrill et al., 1996; Ferrini-Mundy & Graham, 1994; Tall & Vinner, 1981; Williams, 1991). Researchers identified difficulties stemming from a number of issues: the language of limits (Cornu, 1991; Williams, 1991), multiple quantification in the formal definition (Dubinsky, Elderman, & Gong, 1988; Dubinsky & Yiparaki, 2000; Swinyard & Lockwood, 2007), implicit dependencies among quantities in the definition (Roh & Lee, 2011a, 2011b), and persistent notions pertaining to the existence of infinitesimal quantities (Ely, 2010). Limits and continuity are often couched as formalizations of approaching and connectedness respectively. However, the standard, formal definitions display much more subtlety and complexity. That complexity often baffles students who cannot perceive the necessity for so many moving parts. Thus learning the concepts and formal definitions in real analysis are fraught both with need to acquire proficiency with conceptual tools such as quantification and to help students perceive conceptual necessity for these tools. This means students often cannot coordinate their concept image with the concept definition, inhibiting their acculturation to advanced mathematical practice, which emphasizes concept definitions.

See the entire article (note that the online article provides links to the papers cited above).

To summarize, in the field of education, researches decidedly have not come to the conclusion that epsilon, delta definitions are either "simple", "clear", or "common sense". Meanwhile, mathematicians often express contrary sentiments. Two examples are given below.

...one cannot teach the concept of limit without using the epsilon-delta definition. Teaching such ideas intuitively does not make it easier for the student it makes it harder to understand. Bertrand Russell has called the rigorous definition of limit and convergence the greatest achievement of the human intellect in 2000 years! The Greeks were puzzled by paradoxes involving motion; now they all become clear, because we have complete understanding of limits and convergence. Without the proper definition, things are difficult. With the definition, they are simple and clear.

see Kleinfeld, Margaret; Calculus: Reformed or Deformed? Amer. Math. Monthly 103 (1996), no. 3, 230-232.

I always tell my calculus students that mathematics is not esoteric: It is common sense. (Even the notorious epsilon, delta definition of limit is common sense, and moreover is central to the important practical problems of approximation and estimation.)

see Bishop, Errett; Book Review: Elementary calculus. Bull. Amer. Math. Soc. 83 (1977), no. 2, 205--208.

Having presented some published references, I would like to complement the published references with anecdotal evidence drawn from this very page, namely the sentiment that

epsilon-delta definition is immediately appealing to students with a certain clarity of mind, etc.

Such sentiments I believe are common and reveal a belief that these convoluted definitions are "natural" and a failure to follow them possibly constitutes an absence of a "clarity of mind".

When one compares the upbeat assessment found in the mathematics community and the somber assessments common in the education community, sometimes one wonders whether they are talking about the same thing. How does one bridge the gap between the two assessments? Are they perhaps dealing with distinct student populations? Are there perhaps education studies providing more upbeat assessments than Dawkins' article would suggest?

Note 1. A number of editors have commented by now on the subject of alternatives. Thus, Neil Strickland notes that "People may think that rigorous analysis is valuable despite this [teaching difficulty], and that certain other proposed teaching methods are no better, but those are different questions." In response, I would like to mention that the comparison of distinct approaches to teaching analysis is a fascinating subject, on which I have first hand experience and much to say. However, this subject is not that of this question.

Note 2. Evidence from classroom experience comparing the two approaches has been presented in this recent publication based on the opinion poll of the students involved.

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    $\begingroup$ The fact that this circle of ideas can be difficult and discouraging is one of its good features, insofar as it aids in the early identification and weeding-out of students who don't have an aptitude for math. $\endgroup$ Feb 20 '14 at 16:13
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    $\begingroup$ Katz I feel like many of your questions are thinly disguised attempts to push an agenda about infinitesimals, which I don't think is really in the spirit of the site. $\endgroup$
    – arsmath
    Feb 20 '14 at 16:53
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    $\begingroup$ @StevenLandsburg: don't ask me for a teaching reference letter. $\endgroup$
    – Ben McKay
    Feb 20 '14 at 17:12
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    $\begingroup$ I think many of the difficulties with $\epsilon-\delta$ or $\epsilon-N$ analysis disappear when they are properly motivated. How many terms do you need in the Riemann sum to get an approximation with a maximum error of $0.001$? This is question is not esoteric at all: it is essential if you are going to numerically integrate anything in the real world. $\endgroup$ Feb 20 '14 at 18:29
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    $\begingroup$ @katz: what is this "upbeat assessment" of which you speak? I don't think I have heard many people deny that average students find rigorous analysis hard, and that current teaching methods do not succeed in making many of them understand it. People may think that rigorous analysis is valuable despite this, and that certain other proposed teaching methods are no better, but those are different questions. $\endgroup$ Feb 27 '14 at 17:38

I would attribute this to sample bias -- which is "distinct student populations" of a sort.

Mathematicians who teach epsilon-delta definitions rarely study student reception of those definitions in a rigorous way. If they did, they would probably be publishing in education journals, not math journals.

Math teachers evaluating the reception of these ideas probably make informal evaluations, focusing on the most obvious students: those who participated actively in class, those who stayed in touch with their teachers, those who continued with more mathematics. Those students are substantially more likely to appreciate the epsilon-delta definitions.

Education researchers are less likely to suffer from this bias, though surely they suffer from others.

  • $\begingroup$ This is an interesting hypothesis. If you are in touch with some education specialists, could you ask around to see if there are studies that could confirm the idea of "sample bias"? $\endgroup$ Feb 27 '14 at 19:36

I don't teach calculus, but I think what really gives students trouble are nested universal and existential quantifiers --- not epsilons and deltas in particular. I teach these in the language of adversarial games between two players. Usually, "for all" is the adversary, who gets to choose any object, so we must be prepared for "the worst". "There exists" is the "me" player, who must provide some response to the adversary's challenge. Additional levels of nesting correspond to multiple game rounds. I also instruct my students to read quantified formulas from the inside-out: First understand the "last move" in the game and slowly make your way to the beginning. For my small sample size, this seems to be an effective technique.

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    $\begingroup$ I've used this in a Calculus classroom. It seems to be very illuminating to some students, and mean almost nothing to others. All in all, I think it helps enough students that it's worth the time it takes to explain and work out a few examples. $\endgroup$ Jul 17 '17 at 17:51
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    $\begingroup$ The problem I've encountered is that the roles switch once you use continuity rather than proving it: for a function that is assumed continuous, the "for all" player is me, and the "there exists" player is an oracle who supplies the crucial ingredient in my proof. $\endgroup$
    – LSpice
    Dec 4 '17 at 13:33

To bridge that gap, I think that math departments in introductory analysis should be showing and motivating as many alternative definitions of the derivative and the integral as they can. Considering hypothetical or alternate definitions exercises one's intuition muscles, and it helps address some of the common hang-ups that some of the students tend to get caught in. I favor doing epsilon-delta side by side with the infinitesimal approach just so the students can experience the differences in conceptualization.

If time is limited, it is better even to emphasis definition and theorem than the actual proofs themselves. One of the main goals of introductory proof making courses should be to prepare students to ask intelligent and critical questions when they are presented with abstract theoretical propositions, rather than being dependent on an instructor to spoon feed them.

Many physics students enter graduate programs unprepared for conceptualization and intuition because they lack the basic motivation and building blocks for thinking in terms of definition, theorem, and proof. One result has been physics departments filled with instructors who can't even explain the basic justification for manipulations they do on the board by rote. That kind of rote learning results in unnecessary conceptual compartmentalization that can be very detrimental.

I doubt the situation will improve unless the math departments start showing students both epsilon-delta and hyperreal+Sato approaches. Then students would have the maturity coming from thinking of the motivations for using alternate definitions.

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    $\begingroup$ Interesting reply. Feel free to elaborate on Sato. $\endgroup$ Feb 27 '14 at 14:37
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    $\begingroup$ I think it is very smart to explain how the $\epsilon$, $\delta$ definition captures Newton's and Leibniz's informal notion of an infinitessimal, but I think that actually trying to teach real analysis students about hyperreals is a very bad idea. Your choices are either to try to define things properly using non-principal ultrafilters or whatever, in which case $\delta$'s and $\epsilon$'s will look like a walk in the park; or wave your hands at the definition, which defeats the whole purpose of a real analysis class. $\endgroup$ Feb 28 '14 at 12:55
  • $\begingroup$ @PaulSiegel, I don't think anybody would argue that analysis can (or should) be taught without $\epsilon,\delta$ definitions. Rather, we are talking about freshman calculus. Just as you don't construct the real line in freshman calculus, it would be inappropriate to construct the hyperreals. Rather, you define infinitesimals via violation of the Archimedean property, and show students how to work with them rigorously. In my first hand experience, they relate to this much more positively than being dressed to perform multiple-quantifier epsilontic stunts on pretense of being taught calculus $\endgroup$ Mar 4 '14 at 19:49
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    $\begingroup$ @katz: Hold on, now I'm confused. The first sentence in the text of your question referred to "definitions in real analysis", and your quote from Dawkins begins with "Student difficulties with real analysis definitions." Additionally, Jake Shreffler's answer refers to "introductory analysis" classes. The question about what precise definitions should be taught in a freshman calculus class is a very different one (whose best answer is probably "none"), so if that is what you are asking you should make it clearer in your question. $\endgroup$ Mar 4 '14 at 21:04
  • $\begingroup$ @PaulSiegel, Dawkins wrote "both at the calculus and analysis level of instructions". In the title of his section 2.3 he is merely including calculus as part of analysis and using analysis as a general term. Most of the studies he cites have to do with calculus teaching. Kleinfeld's title mentions "calculus" and Bishop is similarly talking about calculus teaching. $\endgroup$ Mar 5 '14 at 14:04

Maybe this is in part due to what we might call Gromov's zero-one law (from this paper on pg. 64, and also quoted in the conclusions here)...

...this common and unfortunate fact of the lack of an adequate presentation of basic ideas and motivations of almost any mathematical theory is, probably, due to the binary nature of mathematical perception: either you have no inkling of an idea or, once you have understood it, this very idea appears so embarrassingly obvious that you feel reluctant to say it aloud; moreover, once your mind switches from the state of darkness to the light, all memory of the dark state is erased and it becomes impossible to conceive the existence of another mind for which the idea appears nonobvious...

Just in case... I'm not saying the math education people don't understand limits :).


I believe that epsilon-delta definition is immediately appealing to students with a certain clarity of mind, who find the heuristic notion of limit vague and confusing. It certainly was the case with me. However, these are a small minority of students in a typical first or second semester calculus course. They may later become math majors and get a serious proof-based education.

At the same time, the majority of students we teach calculus to are NOT going to be mathematicians. It is OK for an engineer to be vague about some foundational notions of mathematics and rigorous definition of a limit is one such notion. We can teach it much the same way a good colloquium or seminar speaker glosses over unpleasant technical details, occasionally at the expense of correctness.

I personally always put the epsilon-delta definition on the board and explain it as best as I can, but with a time cap on the discussion. It is presented as a window into math major experience, rather than a central part of the curriculum. As a middle ground, when I talk about a limit, I make sure to use a catch phrase "wobble subsides" to avoid the common misperception that functions $f(x)$ that have a limit at $x\to a$ are monotone in a small neighborhood of $a$.

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    $\begingroup$ As I note in my comment above, it is exactly engineers who would profit the most from serious attention to $\epsilon-\delta$ in the form of rigorous error analysis. In every application, you need to know how good a control on inputs you need to guarantee a given error tolerance on the outputs. Without this bridges collapse, and airplanes fail to get off the ground. $\endgroup$ Feb 27 '14 at 18:05
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    $\begingroup$ @Steven Gubkin, I agree with your comment about error analysis -- but I would emphasize that you need a uniform $\epsilon-\delta$ analysis for that application. E.g.: What control on the radius do you need to guarantee an error within 1% on the volume? The application is not good with an answer depending on $r$, the actual and unknown radius; it's better to give the answer from uniform continuity that applies to a range of $r$'s. $\endgroup$
    – Matt F.
    Feb 27 '14 at 18:20
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    $\begingroup$ This answer reminds me of how I was taught limits the first time around, which was a kind of middle ground between "total handwaving" and rigorous $\epsilon,\delta$ definitions. Namely I was told, only in words, something along the lines of "$f(x) \to L$ as $x \to a$ if you can make $f(x)$ arbitrarily close to $L$ by taking $x$ arbitrarily close to $a$". This is not 100% unambiguous, but I found it both enlightening and intelligible at the time. $\endgroup$
    – Ian
    Jul 10 '17 at 18:29
  • $\begingroup$ @Ian, that's a useful heuristic. Another useful heuristic is in terms of the "trough/target" dichotomy, which I use when I teach this based on infinitesimals as well. $\endgroup$ Jul 11 '17 at 9:22
  • $\begingroup$ It is common for us to justify the $\epsilon$-$\delta$ explanation of limits in terms of error tolerance in practical situations. However, there is a key difference. Engineers don't need the first quantifier. They have an a priori value of $\epsilon$ in mind and solve for the $\delta$. They do not need to solve this for every possible value of $\epsilon$. $\endgroup$
    – Deane Yang
    Jul 16 '17 at 16:41

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