Proposition: (\epsilon)-(\delta) Definition of Continuity

Let \((X,d_x)\) and \((Y,d_y)\) be metric spaces and let \(f:X\to Y\) be a function \(f\) is continuous at a point \(a\in X\) , if and only if for every \(\epsilon > 0\) there is a \(\delta > 0\) such that \[d_y(f(x),f(a)) < \epsilon\] for all \(x\in X\) with \[d_x(x,a) < \delta.\]

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

Proofs: 1

  1. Definition: Modulus of Continuity of a Continuous Function

Chapters: 1
Definitions: 2 3 4 5
Propositions: 6 7 8

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