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Adding local short description: "Function which is not continuous at any point of its ___domain", overriding Wikidata description "function which is not continuous at any point in the function's ___domain" (Shortdesc helper) |
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{{Short description|Function which is not continuous at any point of its ___domain}}
{{more citations needed|date=September 2012}}
In [[mathematics]], a '''nowhere continuous function''', also called an '''everywhere discontinuous function''', is a [[function (mathematics)|function]] that is not [[continuous function|continuous]] at any point of its [[___domain of a function|___domain]]. If ''f'' is a function from [[real number]]s to real numbers, then ''f'' is nowhere continuous if for each point ''x'' there is an {{nowrap|''ε'' > 0}} such that for each {{nowrap|''δ'' > 0}} we can find a point ''y'' such that {{nowrap|0 < {{abs|''x'' − ''y''}} < ''δ''}} and {{nowrap|{{abs|''f''(''x'') − ''f''(''y'')}} ≥ ''ε''}}. Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.
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