Nowhere continuous function: Difference between revisions

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Revision as of 11:00, 2 May 2004

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One example of such a function is a function <i>f</i> on the [[real number|real numbers]] such that <i>f(x)</i> is 1 if <i>x</i> is a [[rational number]], but 0 if <i>x</i> is not rational. If we look at this function in the vicinity of some number <i>y</i>, there are two cases:

If y is rational, then <i>f(y)</i>=1. To show the function is not continuous at y, we need find a single <i>&epsilon;</i> which works in the above definition. In fact, 1/2 is such an <i>&epsilon;</i>, since we can find an irrational number <i>z</i>arbitrarily close to y and <i>f(z)</i>=0, at least 1/2 away from 1. If y is irrational, then <i>f(y)</i>=0. Again, we can take <i>&epsilon;</i>=1/2, and this time we pick <i>z</i> to be an rational number as close to <i>y</i> as is required. Again, <i>f(z)</i> is more than 1/2 away from <i>f(y)</i>

The discontinuities in this function occur because both the rational and irrational numbers are [[dense]] in the [[real number]]s. It was originally investigated by [[Johann_Peter_Gustav_Lejeune_Dirichlet|Dirichlet]].)

== External link ==

* [http://mathworld.wolfram.com/DirichletFunction.html Dirichlet Function -- from MathWorld]