Nowhere continuous function: Difference between revisions

Revision as of 08:04, 11 November 2003

Revision as of 02:32, 11 January 2004

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A '''nowhere continuous''' [[function]] is (tautologically) a function that is not [[Continuouscontinuous]] at any point. That is to say, <i>f(x)</i> is nowhere continuous for each point <i>x</i> there is an <i>&epsilon; >0</i> such that for each <i>&delta; >0</i> we can find a point <i>y</i> such that <i>|x-y|<&delta; </i> and <i>|f(x)-f(y)|>&epsilon; </i>. Basically, this is a statement that at each point we can choose a distance such that points arbitrarily close to our original point are taken at least that distance away.

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

On 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 Thiswe satisfieslook theat abovethis definitionfunction within <i>&epsilon;</i>the =1/2vincinity forof eachsome number <i>xy</i>, because both the rational and irrational numbersthere are [[dense]] in the [[real number]]s. (This example is duetwo tocases: [[Johann_Peter_Gustav_Lejeune_Dirichlet|Dirichlet]].)

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]].)