Both education researchers and mathematicians discuss the challenge of (epsilon, delta) type definitions in real analysis and the student reception of them. My impression has been that mathematicians often hold an upbeat opinion on the success of student reception of this, whereas education researchers often stress difficulties and their "baffling" and "inhibitive" effect (see below).
Paul Dawkins recently expressed a typical educational perspective on this:
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. Researchers identified difficulties stemming from a number of issues: the language of limits, multiple quantification in the formal definition, implicit dependencies among quantities in the definition, and persistent notions pertaining to the existence of infinitesimal quantities. 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; the online article provides links to references for several of the comments 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 I compare the upbeat assessment common in the mathematics community and the somber assessments common in the education community, sometimes I wonder 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?
There is a parallel thread with a bounty offered at https://math.stackexchange.com/questions/683513/evaluating-the-reception-of-epsilon-delta-definitions so you can respond there (or here if you prefer).
Note 1. Several 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.
(c) misguided educators: educators are not sufficiently concerned with rigor and cause horrible confusion to the students.
Most of these approaches 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.