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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 anyone 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 use it, and when is, according to the standards today, the correct setting 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 at 10:58
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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. – Andrew Stacey Feb 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 at 12:19
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@wasteofspace The character limit on comments meant I missed off the crucial last words: "... hand written by mathematicians.". – Andrew Stacey Feb 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 at 23:04

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Historically there has been a lot of confusion over the two forms, At one point Unicode swapped the reference glyphs to add to the confusion. I added a special section about epsilon to the XML/HTML entities spec

http://www.w3.org/2003/entities/2007doc/#epsilon

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.

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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. – tohecz Feb 13 at 12:23

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