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Interesting Example
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.
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.
== Interesting examples ==
The [[characteristic function]] of <math>\mathbb{Q}\,</math> in <math>\mathbb{R}\,</math>, <math>\chi_{\mathbb{Q}} : \mathbb{R} \rightarrow \mathbb{R}</math>, defined as
<math>\chi_{\mathbb{Q}}(x)</math><math> = \left\{\begin{matrix}
\,1 \quad &: &\quad x \in \mathbb{Q} \\
\,0 \quad &: &\quad x \in \mathbb{R} \setminus \mathbb{Q}
\end{matrix}\right.</math>
is nowhere continuous.
[[category: mathematics]]
[[category: mathematics]]
[[category: Topology]]
[[category: Topology]]