Revision as of 16:24, 19 August 2004 edit Michael Hardy (talk \| contribs) Administrators 210,597 edits No edit summary ← Previous edit		Revision as of 16:44, 19 August 2004 edit undo Michael Hardy (talk \| contribs) Administrators 210,597 edits No edit summary Next edit →
Line 1: In [[mathematics]], a '''nowhere continuous''' [[function (mathematics)\|function]], also called an '''everywhere discontinuous''' function, is a function that is not [[continuous]] at any point of its [[~~function~~ ___domain (mathematics)\|___domain]]. If ''f'' is a function from [[real number]]s to real numbers, then ''f''(''x'') is nowhere continuous if for each point ''x'' there is an ''ε''>0 such that for each ''δ''>0 we can find a point ''y'' such that \|''x''−''y''\|<''δ'' and \|''f''(''x'')−''f''(''y'')\|≥''ε''. The import of this statement is that no matter how close we get to any fixed point, there are nearby points at which the function takes not-nearby values. More general definitions of this kind of function can be obtained, by replacing the [[absolute value]] by the distance function in a [[metric space]], or the continuity definition by the definition of continuity in a [[topological space]].

Nowhere continuous function: Difference between revisions