In [[mathematics]], a '''nowhere continuous function''' [[function (mathematics)|function]], also called an '''everywhere discontinuous function''' function, is a [[function (mathematics)|function]] that is not [[continuous function|continuous]] at any point of its [[___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]].
