Editing Projectively extended real line (section)

==== Extended definition of limits ====
Let <math>A \subseteq \widehat{\mathbb{R}}</math>. Then ''p'' is a [[limit point]] of ''A'' if and only if every neighbourhood of ''p'' includes a point <math>y \in A</math> such that <math>y \neq p.</math>

Let <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}}, A \subseteq \widehat{\mathbb{R}}, L \in \widehat{\mathbb{R}}, p \in \widehat{\mathbb{R}}</math>, ''p'' a limit point of ''A''. The limit of ''f''{{hairsp}}(''x'') as ''x'' approaches ''p'' through ''A'' is ''L'', if and only if for every neighbourhood ''B'' of ''L'', there is a punctured neighbourhood ''C'' of ''p'', such that <math>x \in A \cap C</math> implies <math>f(x) \in B.</math>

This corresponds to the regular [[continuity (topology)|topological definition of continuity]], applied to the [[subspace topology]] on <math>A\cup \lbrace p \rbrace,</math> and the [[restriction (mathematics)|restriction]] of ''f'' to <math>A \cup \lbrace p \rbrace.</math>