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→More precise definition: ε-δ-definition for metric spaces
A continuous function maps a [[compact space]] to a [[compact space]].
A continuous function maps a [[compact space]] to a [[compact space]].
=== Metric Spaces ===
Let <math>X\,</math> and <math>Y\,</math> be to [[metric space]]s, and <math>f: X \rightarrow Y</math> a function between these two sets. Then <math>f\,</math> is ''continuous'' in <math>x_0 \in X</math> if for all <math>\epsilon > 0\,</math> there is a <math>\delta > 0\,</math> such that for all <math>x\,</math> with
::<math> |x - x_0| < \delta\,</math>
we have
::<math>|f(x) - f(x_0)| < \epsilon \,</math>.
This is the notorious <math>\epsilon-\delta-\,</math>''definition'' of continuity. Especially, it works for the metric spaces <math>\mathbb{R}\,</math> and <math>\mathbb{R}^n\,</math>, and it is used in any college level course on calculus.
[[category: mathematics]]
[[category: mathematics]]
[[category: Topology]]
[[category: Topology]]