Definition: Limit of a Function

Let \((X,d)\) and \((Y,d)\) be metric spaces and let \(T\subseteq X\) be a subset with a boundary point \(b\in X\). Furthermore, let \(f\colon T\longrightarrow Y\,\) be a function. The point \(a\in Y\) is called the limit of \(f\) at the point \(b\), denoted by \[\operatorname {lim} _{x\rightarrow a}\,f(x)=b\]

if for each \(\epsilon > 0\) there exists a \(\delta > 0\) such that for each \(x\) contained in the neighborhood of \(a\in T\), \(f(x)\) is contained in the neighborhood of \(b\in Y\), formally

\[\forall\,\epsilon > 0\,\exists\delta > 0:\,x\in T\cap B\left(a,\delta \right)\Longrightarrow f(x)\in Y\cap B\left(b,\epsilon \right).\]

Definitions: 1 2

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Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück