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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 \(a\in X\).
Table of Contents
Proofs: 1
- Definition: Modulus of Continuity of a Continuous Function
Mentioned in:
Chapters: 1
Definitions: 2 3 4 5
Propositions: 6 7 8
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References
Bibliography
- Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984
