Nowhere continuous function: Difference between revisions

Revision as of 11:01, 5 June 2017

Revision as of 17:56, 6 September 2017

Line 1:

{{refimprove|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''(''x'') 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'' &minus; ''y''}} < ''δ''}} and {{nowrap|{{abs|''f''(''x'') &minus; ''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.

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 by using the definition of continuity in a [[topological space]].