Nowhere continuous function: Difference between revisions

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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 ε&nbsp;&gt;&nbsp;0 such that for each δ&nbsp;&gt;&nbsp;0 we can find a point ''y'' such that 0< |''x''&nbsp;&minus;&nbsp;''y''|&nbsp;&lt;&nbsp;δ and |''f''(''x'')&nbsp;&minus;&nbsp;''f''(''y'')|&nbsp;≥&nbsp;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]].