Nowhere continuous function: Difference between revisions

Revision as of 10:12, 10 March 2004

Revision as of 16:58, 22 March 2004

Line 1:

In [[mathematics]], a '''nowhere continuous''' [[function (mathematics)|function]] is a function that is not [[continuous]] at any point of its [[function ___domain|___domain]]. Suppose that ''f'' is a function from [[real number]]s to real numbers. Then, <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>. The import of this statement is 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 from the function's value.

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]].