Definition: Continuous Functions in Metric Spaces

Let \((X,d)\) and \((Y,d)\) be two metric spaces and \(f:X\to Y\) be a function \(f\) is continuous at a point \(a\in X\), if

\[\lim_{x\to a} f(x)=f(a),\] i.e. if for each convergent sequence \((x_n)_{n\in\mathbb N}\) of points \(x_n\in X\) with \(\lim x_n=a\) we have \(\lim f(x_n)=f(a)\).

The function \(f\) is called continuous on \(X\), if \(f\) is continuous in every point \(a\in X\).

  1. Definition: Pointwise and Uniform Convergence
  2. Proposition: (\epsilon)-(\delta) Definition of Continuity

Corollaries: 1
Definitions: 2 3 4 5
Parts: 6
Proofs: 7 8 9 10 11 12
Propositions: 13 14
Theorems: 15 16

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  1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984