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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'' 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.
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]].
[[File:Dirichlet beta function plot.png|thumb|right|Plot of Dirichlet beta-function of real argument]]▼
==Dirichlet function==
▲[[File:Dirichlet beta function plot.png|thumb|right|Plot of Dirichlet beta-function of real argument]]
One example of such a function is the [[indicator function]] of the [[rational number]]s, also known as the '''Dirichlet function''', named after German mathematician [[Peter Gustav Lejeune Dirichlet]].<ref>Lejeune Dirichlet, P. G. (1829) "Sur la convergence des séries trigonométriques qui servent à répresenter une fonction arbitraire entre des limites donées" [On the convergence of trigonometric series which serve to represent an arbitrary function between given limits], ''Journal für reine und angewandte Mathematik'' [Journal for pure and applied mathematics (also known as ''Crelle's Journal'')], vol. 4, pages 157–169.</ref> This function is denoted as ''I''<sub>'''Q'''</sub> and has [[___domain of a function|___domain]] and [[codomain]] both equal to the [[real number]]s. ''I''<sub>'''Q'''</sub>(''x'') equals 1 if ''x'' is a [[rational number]] and 0 if ''x'' is not rational. If we look at this function in the vicinity of some number ''y'', there are two cases:
*If ''y'' is rational, then {{nowrap|1=''f''(''y'') = 1}}. To show the function is not continuous at ''y'', we need to find an ''ε'' such that no matter how small we choose ''δ'', there will be points ''z'' within ''δ'' of ''y'' such that ''f''(''z'') is not within ''ε'' of {{nowrap|1=''f''(''y'') = 1}}. In fact, 1/2 is such an ''ε''. Because the [[irrational number]]s are [[dense set|dense]] in the reals, no matter what ''δ'' we choose we can always find an irrational ''z'' within ''δ'' of ''y'', and {{nowrap|1=''f''(''z'') = 0}} is at least 1/2 away from 1.
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