Nowhere continuous function: Difference between revisions

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==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>{{cite journal |first=Lejeune Dirichlet, |last=P. G. (1829) "|title=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=Journal für reine und angewandte Mathematik'' [Journal for pure and applied mathematics (also known as ''Crelle's Journal'')], vol. |volume=4, |pages=157–169 157–169|year=1829 |url=https://eudml.org/doc/183134}}</ref> This function is denoted as ''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'') = 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.

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