Nowhere continuous function: Difference between revisions

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{{Short description|Function which is not continuous at any point of its ___domain}}

{{more citations needed|date=September 2012}}

In [[mathematics]], a '''nowhere continuous function''', also called an '''everywhere discontinuous function''', is a [[function (mathematics)|function]] that is not [[continuous function|continuous]] at any point of its [[___domain of a function|___domain]]. If <math>f</math> is a function from [[real number]]s to real numbers, then <math>f</math> is nowhere continuous if for each point <math>x</math> there is some <math>\epsilonvarepsilon > 0</math> such that for every <math>\delta > 0,</math> we can find a point <math>y</math> such that <math>|x - y| < \delta</math> and <math>|f(x) - f(y)| \geq \epsilon.varepsilon</math>. Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.

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