$\begingroup$ 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. $\endgroup$

$$\le |(x - x_{0})|^{2} + |2x_{0}(x-x_{0})| < \epsilon$$ by the triangle inequality. But we don't know this is less than epsilon since we now have something that is greater than what we actually know is less than epsilon, this comes up time and time again in many proofs and I can't get my head around why we say this is still less than epsilon.

The backwards epsilon notation for "such that" was introduced by Peano in 1898, e.g. from Jeff Miller's Earliest Uses of Various Mathematical Symbols: Such that. According to Julio González Cabillón, Peano introduced the backwards lower-case epsilon for "such that" in "Formulaire de Mathematiques vol. II, #2" (p. iv, 1898).

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.

Jun 21, 2018 · The epsilon algebra is a simple method compared to the more advanced interval arithmetic which can be used to put bounds on rounding and measurement errors in computation. The advantage of epsilon algebra is that simple rules of calculation are sufficient compared to the complicated calculation rules of interval arithmetic.

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.

Sep 28, 2013 · $\begingroup$ Well, one reason to write longer is to make clear what you understand. People on M.SE have a lot of different experience levels, so when you skip crucial stuff or use odd notation, we have to guess if you understand it.

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为了验证这一感觉，我查阅了W3C标准对epsilon各种变体的详细分类和名称解释，列出下表： 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 …