There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. My impression has been that the mathematical community often holds an upbeat opinion on the success of student reception of this, 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 http://dx.doi.org/10.1016/j.jmathb.2013.10.002 for the entire article (note that the online article provides links to the papers cited above).
To summarize, in the field of education, researchers 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.)
When one compares the upbeat assessment common 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. See also https://mathoverflow.net/questions/158145/assessing-effectiveness-of-epsilon-delta-definitions
Note 2. Two approaches have been proposed to account for this difference of perception between the education community and the math community: (a) sample bias: mathematicians tend to base their appraisal of the effectiveness of these definitions in terms of the most active students in their classes, which are often the best students; (b) student/professor gap: mathematicians base their appraisal on their own scientific appreciation of these definitions as the "right" ones, arrived at after a considerable investment of time and removed from the original experience of actually learning those definitions. Both of these sound plausible, but it would be instructive to have field research in support of these approaches.
We recently published an article reporting the result of student polling concerning the comparative educational merits of epsilon-delta definitions and infinitesimal definitions of key concepts like continuity and convergence, with students favoring the infinitesimal definitions by large margins.
Best Answer
My feeling is that the biggest problem with the epsilon-delta definition is that this is the first time students have ever seen the universal and existential quantifiers. By the time you say, "For every epsilon there exists a delta," you have already lost 95% of your audience before you even get to the business end of the proposition.
And of course the other problem is with the lower-case Greek letters. Students have been seeing x, y, z, and t all their lives; and out of nowhere you show them epsilon and delta.
In other words it's the basic form of the definition that's intimidating and confusing to students; not so much the actual idea, which is simply that you can arbitrarily constrain the output by suitably constraining the input.
Perhaps if instructors started with the conceptual understanding and then spent time explaining "for all" and "there exists" and giving them a gentle introduction to Greek letters used as variables, things would get better.