Nowhere continuous function: Difference between revisions

Revision as of 08:03, 11 November 2003

Revision as of 08:04, 11 November 2003

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A '''nowhere continuous''' [[function]] is (tautologically) a function that is not [[Continuous]] 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 <mathi>\abs{|x-y}|<\&delta; </mathi> 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]].