Nowhere continuous function: Difference between revisions

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Revision as of 17:02, 10 February 2007

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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 the continuity definition by the definition of continuity in a [[topological space]].

One example of such a function is the [[indicator function]] of the [[rational number]]s, also known as the ''[[Dirichlet function]]''. This function is written ''I''_'''Q''' and has [[___domain (mathematics)|___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'') = 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.