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I cannot remember anyone writing the letter epsilon in any other way than \varepsilon in any math class; but in LaTeX \epsilon and \varepsilon are different symbols. Do any of you know why there are two different symbols? (I.e. if \epsilon is the correct way to write the letter epsilon, why aren't mathematicians using it, and when is, according to the standards today, the correct situation to use each of the symbols?)

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Here in TeX.SE what you write between $$ doesn't show as math symbols. But you can mark your code with backticks. I don't know the answer, but I use \epsilon for the Levi-Civita tensor, and \varepsilon for everything else (of course, renaming both commands). – Manuel Feb 13 '13 at 10:58
In print it is easy to distinguish between ϵ and ∈ but when hand written it is harder, so there is a tendency to write ε to make things clear. But I suspect I'm not alone in reading most symbols as "squiggle", "new squiggle", and "squiggle that was used three pages ago for something that I no longer remember" so the actual form is not particularly important. What is important is to make it easy to distinguish between this squiggle and that squiggle. If you think epsilon is confusing, it takes years of practise to be able to distinguish ζ and ξ when hand written. – Loop Space Feb 13 '13 at 11:19
@andrewstacey “it takes years of practise to be able to distinguish ζ and ξ when hand written” isn't entirely true: after a couple of weeks of learning greek, i was writing them quite fluently ... ;-) – wasteofspace Feb 13 '13 at 12:19
@wasteofspace The character limit on comments meant I missed off the crucial last words: "... hand written by mathematicians.". – Loop Space Feb 13 '13 at 12:21
@wasteofspace -- if you want an even more potentially confusing pair, consider times roman v and nu. those probably are distinguishable handwritten, but in print, yech! – barbara beeton Feb 13 '13 at 23:04

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Historically there has been a lot of confusion over the two forms, (the situation with \phi and \varphi is similar but even more confused as at one point Unicode swapped the reference glyphs). I added a special section about epsilon to the XML/HTML entities spec

The situation in TeX is no different really, different communities used different forms of epsilon and it is rather arbitrary which one gets which name. Unicode (now) calls the curly epsilon "GREEK SMALL LETTER EPSILON" (ε) (this is a textual Greek letter rather than a math alphabetic symbol) and the symbol that TeX traditionally assigns to \epsilon is called GREEK LUNATE EPSILON SYMBOL (ϵ) the "symbol" being a hint that this is intended as a mathematical character rather than a textual Greek letter.

From Wikipedia:

The lowercase version has two typographical variants, both inherited from medieval Greek handwriting. One, the most common in modern typography and inherited from medieval minuscule, looks like a reversed "3". The other, also known as lunate or uncial epsilon and inherited from earlier uncial writing, looks like a semicircle crossed by a horizontal bar. While in normal typography these are just alternative font variants, they may have different meanings as mathematical symbols. Computer systems therefore offer distinct encodings for them. In Unicode, the character U+03F5 "Greek lunate epsilon symbol" (ϵ) is provided specifically for the lunate form. In TeX, \epsilon (ϵ) denotes the lunate form, while \varepsilon (ε) denotes the inverted-3 form.

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It is necessary to remark that there are communities that use both, but for different things! (like my community /blushes). This is of course quite confusing and very close to being invalid as a notation. – yo' Feb 13 '13 at 12:23
as noted by @tohecz, some communities use both, for different things. one such use i'm familiar with is that some mathematicians insist on using the lunate epsilon instead of \in. getting that mixed up in an e-document would be really unfortunate. – barbara beeton Jan 14 '14 at 17:39
In my copy of Kelley's “General Topology”, ‘ɛ’ (not lunate) is used for “belongs to”. I think the lunate one is used in the book for cases like “let ϵ > 0”, which is not so good a notation, in my opinion. That's why the vast majority of mathematicians use a clearly distinct variant ‘∈’ of the lunate epsilon for denoting “belongs to” (bigger, wider and centered to the formula axis). – egreg Jan 14 '14 at 18:26

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