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注意到\epsilon的Unicode全名是「Greek Small Letter Epsilon」，即「小写希腊字母Epsilon」；而\varepsilon的Unicode全名则是「Greek Lunate Epsilon Symbol」，即「半月形希腊标志Epsilon」，并没有提到「字母」。

$\begingroup$ Historically, the symbol $\in$ is derived from $\epsilon$, thus it is not impossible to confuse both symbols. Also, not as ubiquitous as its primary usage, this Greek symbol $\epsilon$ or $\varepsilon$ is also used to denote the sign, including Levi-Civita symbol in physics and random sign in probability to name a few. $\endgroup$

Apr 19, 2022 · $\begingroup$ As an aside, the $\in$ takes its basic shape from the Greek letter $\epsilon$. The symbol $\in$ means "is an element of", so using the Greek letter that starts the word "element" as a design inspiration isn't entirely unreasonable. $\endgroup$

Nov 3, 2016 · When adding a "small" positive number $\epsilon$ to some quantity, one should/might have some criterion in mind that $\epsilon$ should be "small enough" so that some properties are satisfied. If one really wants to be precise, than one might want to state explicitly those properties out.

Nov 18, 2021 · In the game our opponents value of epsilon and our responding value of delta must always be strictly greater than zero. Saying "$\epsilon = 0$" or "$\delta = 0$" isn't an allowed move in the game; Our opponents aren't going to win by throwing larger values of $\epsilon$ at us because we can just repeat our previous $\delta$ and win that round

Jun 21, 2016 · The point is that given any $\epsilon > 0$ you have make $\lvert f(x) - f(x_0)\rvert$ smaller than this randomly given $\epsilon$, by choosing/(demonstrating the existence of) a $\delta > 0$ so that when $\lvert x - x_0\rvert$ is smaller than $\delta$, then $\lvert f(x) - f(x_0)\rvert$ is smaller than $\epsilon$. Let me do a simple example.

This is the membership relation, but in set theory this is also known as the epsilon relation, and historically the notation was indeed $\varepsilon$. (For example, I have the book from 1948 by Tarski and Jonsson Cardinal Algebras where such notation is employed.) According to this page it was Peano who used epsilon. I suppose somewhere around ...

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$\begingroup$ I find the presentation of the $\epsilon$-$\delta$ definition as a game tends to help beginning students: the players are Paul the prover, and Alice the adversary.

The epsilon-delta definition of continuity is a fundamental concept in real analysis, explaining how limits and continuity are defined using precise mathematical criteria.