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 (mathematics)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 > 0 such that for each δ > 0 we can find a point ''y'' such that |''x'' − ''y''| < δ and |''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]].

==Dirichlet function==

One example of such a function is the [[indicator function]] of the [[rational number]]s, also known as the '''Dirichlet function''', named after [[Germany|German]] [[mathematician]] [[Peter Gustav Lejeune Dirichlet]].<ref>Dirichlet, J.P.G. Lejeune (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 written ''I''_'''Q''' and has [[___domain (mathematics)of a function|___domain]] and [[codomain]] both equal to the [[real number]]s. ''I''_'''Q'''(''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 ''f''(''y'')&nbsp;=&nbsp;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 ''f''(''y'')&nbsp;=&nbsp;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 ''f''(''z'')&nbsp;=&nbsp;0 is at least 1/2 away from 1.

*If ''y'' is irrational, then ''f''(''y'')&nbsp;=&nbsp;0. Again, we can take ε&nbsp;=&nbsp;1/2, and this time, because the rational numbers are dense in the reals, we can pick ''z'' to be a rational number as close to ''y'' as is required. Again, ''f''(''z'')&nbsp;=&nbsp;1 is more than 1/2 away from ''f''(''y'')&nbsp;=&nbsp;0.