|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{refimprove|date=September 2012}}
| | Person who wrote the articles is called Eusebio. South Carolina is [http://Michaelsbirth.org/ michael's birth] place. The favourite hobby for him and as well his kids is to assist you fish and he's resulted in being doing it for quite some time. Filing has been his profession as news got around. Go to his website track down out more: http://circuspartypanama.com<br><br> |
| 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 > 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]].
| | Feel free to surf to my webpage: clash of clans cheats ([http://circuspartypanama.com http://circuspartypanama.com]) |
| | |
| ==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 - 169.</ref> This function is written ''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 ''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 ''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 ''f''(''z'') = 0 is at least 1/2 away from 1.
| |
| *If ''y'' is irrational, then ''f''(''y'') = 0. Again, we can take ε = 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'') = 1 is more than 1/2 away from ''f''(''y'') = 0.
| |
| In plainer 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)^{2j}\right)\right)</math>
| |
| | |
| 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
| |
| | last = Dunham
| |
| | first = William
| |
| | title = The Calculus Gallery
| |
| | publisher = Princeton University Press
| |
| | date = 2005
| |
| | pages = 197
| |
| | isbn = 0-691-09565-5 }}</ref>
| |
| | |
| In general, if ''E'' is any subset of a [[topological space]] ''X'' such that both ''E'' and the complement of ''E'' are dense in ''X'', then the real-valued function which takes the value 1 on ''E'' and 0 on the complement of ''E'' will be nowhere continuous. Functions of this type were originally investigated by [[Peter Gustav Lejeune Dirichlet]].
| |
| | |
| ==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 ''f(x)-f(y)'' is appreciable (i.e., not [[infinitesimal]]).
| |
| | |
| ==See also==
| |
| *[[Thomae%27s function]] (also known as the popcorn function) — a function that is continuous at all irrational numbers and discontinuous at all rational numbers.
| |
| | |
| ==References==
| |
| <references />
| |
| | |
| ==External links==
| |
| * {{springer|title=Dirichlet-function|id=p/d032860}}
| |
| * [http://mathworld.wolfram.com/DirichletFunction.html Dirichlet Function — from MathWorld]
| |
| * [http://demonstrations.wolfram.com/TheModifiedDirichletFunction/ The Modified Dirichlet Function] by George Beck, [[The Wolfram Demonstrations Project]].
| |
| | |
| [[Category:Topology]]
| |
| [[Category:Mathematical analysis]]
| |
| [[Category:Types of functions]]
| |
Person who wrote the articles is called Eusebio. South Carolina is michael's birth place. The favourite hobby for him and as well his kids is to assist you fish and he's resulted in being doing it for quite some time. Filing has been his profession as news got around. Go to his website track down out more: http://circuspartypanama.com
Feel free to surf to my webpage: clash of clans cheats (http://circuspartypanama.com)