Nowhere continuous function: Difference between revisions

Revision as of 16:44, 19 August 2004

Revision as of 23:29, 25 August 2004

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*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 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 [[Johann_Peter_Gustav_Lejeune_DirichletJohann Peter Gustav Lejeune Dirichlet|Dirichlet]].

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