non-uniformly continuous function

We assert that the real functionxsin1x  is not uniformly continuousPlanetmathPlanetmath on the open intervalDlmfPlanetmath(0, 1).

For proving this, we make the antithesis that there exists a positive number δ such that

|f(x1)-f(x2)|< 1always whenx1,x2(0, 1)and|x1-x2|<δ.



where the integer n is so great that  x1<δ2,  x2<δ2.  Then we have



f(x1)-f(x2)= 1-(-1)= 2.

This contradictory result shows that the antithesis is wrong.

Title non-uniformly continuous function
Canonical name NonuniformlyContinuousFunction
Date of creation 2013-03-22 19:00:07
Last modified on 2013-03-22 19:00:07
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Example
Classification msc 26A15
Related topic PointPreventingUniformConvergence
Related topic ReductioAdAbsurdum