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 {{nowrap|''ε&nbsp;&gt;&nbsp;'' > 0}} such that for each {{nowrap|''δ&nbsp;&gt;&nbsp;'' > 0}} we can find a point ''y'' such that {{nowrap|0 < {{abs|''x''&nbsp; &minus;&nbsp; ''y''|&nbsp;&lt;&nbsp;}} < ''δ''}} and {{nowrap|{{abs|''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]].

==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 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 - 169157–169.</ref> This function is written ''I''_'''Q''' and has [[___domain 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 {{nowrap|1=''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 {{nowrap|1=''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 {{nowrap|1=''f''(''z'')&nbsp; =&nbsp; 0}} is at least 1/2 away from 1.

*If ''y'' is irrational, then {{nowrap|1=''f''(''y'')&nbsp; =&nbsp; 0}}. Again, we can take {{nowrap|1=''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, {{nowrap|1=''f''(''z'')&nbsp; =&nbsp; 1}} is more than 1/2 away from {{nowrap|1=''f''(''y'')&nbsp; =&nbsp; 0}}.

In less rigorous terms, between any two irrationals, there is a rational, and vice versa.

The ''Dirichlet function'' can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:

:<math>f(x)=\lim_{k\to\infty}\left(\lim_{j\to\infty}\left(\cos(k!\pi x)\right)^{2j}\right)</math>

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for integer ''j'' and ''k''.

This shows that the ''Dirichlet function'' is a [[Baire function|Baire class]] 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a [[meagre set]].<ref>{{cite book

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==Hyperreal characterisation==

A real function ''f'' is nowhere continuous if its natural [[Hyperreal number|hyperreal]] extension has the property that every ''x'' is infinitely close to a ''y'' such that the difference {{nowrap|''f''(''x'')- − ''f''(''y)'')}} is appreciable (i.e., not [[infinitesimal]]).

==See also==

*[[Thomae%27s's function]] (also known as the popcorn function) &mdash; a function that is continuous at all irrational numbers and discontinuous at all rational numbers.

==References==