TeXipedia

Epsilon

Represents a capital epsilon letter commonly used in mathematical notation, particularly in set theory and analysis.

Overview

Serves as an important mathematical symbol across multiple domains, particularly in advanced mathematics and theoretical physics.

  • Frequently used to denote small positive quantities or error terms in analysis
  • Common in set theory to represent membership relations
  • Appears in mathematical proofs and limit definitions
  • Often used in conjunction with delta in epsilon-delta proofs of continuity and limits
  • Important in mathematical optimization and algorithm analysis for expressing bounds

Examples

Using capital Epsilon in set theory to denote a small positive real number.

∣x−y∣<E  ⟹  f(x)≈f(y)|x - y| < \Epsilon \implies f(x) \approx f(y) 
|x - y| < \Epsilon \implies f(x) \approx f(y)

Defining an epsilon-neighborhood in topology.

BE(x0)={x∈X:d(x,x0)<E}B_\Epsilon(x_0) = \{x \in X : d(x,x_0) < \Epsilon\} 
B_\Epsilon(x_0) = \{x \in X : d(x,x_0) < \Epsilon\}

Expressing the epsilon-delta definition of a limit.

∀E>0,∃δ>0:∣x−a∣<δ  ⟹  ∣f(x)−L∣<E\forall \Epsilon > 0, \exists \delta > 0 : |x-a| < \delta \implies |f(x)-L| < \Epsilon 
\forall \Epsilon > 0, \exists \delta > 0 : |x-a| < \delta \implies |f(x)-L| < \Epsilon