example of jump discontinuity

The elementary (http://planetmath.org/ElementaryFunction) real function


has a jump discontinuity at the origin, since



  • if  x0-,  then  1x-,  e1x0,  11+e1x1;

  • if  x0+,  then  1x,  e1x,  11+e1x0.

These results can be seen also from the series of the function gotten by performing the divisions:  for  x<0  we obtain the converging (http://planetmath.org/ConvergePlanetmathPlanetmath) alternating seriesMathworldPlanetmath (http://planetmath.org/LeibnizEstimateForAlternatingSeries)


and for  x>0  the series


Note.  The derivativePlanetmathPlanetmath of the function may be written as


and thus we have the one-sided limitslimx0±f(x)=0 (see growth of exponential function).

Figure 1: Graph of the function f with jump discontinuity
Title example of jump discontinuity
Canonical name ExampleOfJumpDiscontinuity
Date of creation 2013-03-22 16:25:02
Last modified on 2013-03-22 16:25:02
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Example
Classification msc 26A15
Classification msc 54C05
Related topic ExponentialFunction
Related topic ImproperLimits