Limit of a function: Difference between revisions

Apart from open intervals, limits can be defined for functions on arbitrary subsets of '''R''', as follows {{harv|Bartle|Sherbert|2000}}. Let ''f'' be a real-valued function defined on a subset ''S'' of the real line. Let ''p'' be a [[limit point]] of ''S''&mdash;that is, ''p'' is the limit of some sequence of elements of ''S'' distinct from p. The limit of ''f'', as ''x'' approaches ''p'' from values in ''S'', is ''L'' if, for every {{nowrap|''ε'' > ''0''}}, there exists a {{nowrap|''δ'' > ''0''}} such that {{nowrap|0 < {{abs|''x'' − ''p''}} < ''δ''}} and {{nowrap|''x'' ∈ ''S''}} implies {{nowrap|{{abs|''f''(''x'') − ''L''}} < ''ε''}}.