You are here
Homeexample of jump discontinuity
Primary tabs
example of jump discontinuity
$f\colon\,x\mapsto\frac{1}{1+e^{\frac{1}{x}}}$ 
has a jump discontinuity at the origin, since
$\lim_{{x\to 0}}f(x)=1\quad\mathrm{and}\quad\lim_{{x\to 0+}}f(x)=0.$ 
Indeed,

if $x\to 0$, then $\displaystyle\frac{1}{x}\to\infty$, $\displaystyle e^{\frac{1}{x}}\to 0$, $\displaystyle\frac{1}{1+e^{\frac{1}{x}}}\to 1$;

if $x\to 0+$, then $\displaystyle\frac{1}{x}\to\infty$, $\displaystyle e^{\frac{1}{x}}\to\infty$, $\displaystyle\frac{1}{1+e^{\frac{1}{x}}}\to 0$.
These results can be seen also from the series expansions of the function gotten by performing the divisions: for $x<0$ we obtain the converging alternating series
$\displaystyle 1:(1+e^{{\frac{1}{x}}})=\sum_{{k=0}}^{\infty}(1)^{k}e^{{\frac{k% }{x}}}=1e^{{\frac{1}{x}}}+e^{{\frac{2}{x}}}e^{{\frac{3}{x}}}+\ldots$ 
and for $x>0$ the series
$\displaystyle 1:(e^{{\frac{1}{x}}}+1)=\sum_{{k=1}}^{\infty}(1)^{{k+1}}e^{{% \frac{k}{x}}}=e^{{\frac{1}{x}}}e^{{\frac{2}{x}}}+e^{{\frac{3}{x}}}+\ldots$ 
Note. The derivative of the function may be written as
$f^{{\prime}}(x)=\frac{1}{x^{2}(e^{{\frac{1}{x}}}+1)(1+e^{\frac{1}{x}})},$ 
and thus we have the onesided limits $\displaystyle\lim_{{x\to 0\pm}}f^{{\prime}}(x)=0$ (see growth of exponential function).
Mathematics Subject Classification
26A15 no label found54C05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis
Apr 20
new image: informationtheoreticdistributedmeasurementdds.png by rspuzio
new image: informationtheoreticdistributedmeasurement4.2 by rspuzio
new image: informationtheoreticdistributedmeasurement4.1 by rspuzio
new image: informationtheoreticdistributedmeasurement3.2 by rspuzio
new image: informationtheoreticdistributedmeasurement3.1 by rspuzio
new image: informationtheoreticdistributedmeasurement2.1 by rspuzio
Apr 19
new collection: On the InformationTheoretic Structure of Distributed Measurements by rspuzio
Apr 15
new question: Prove a formula is part of the Gentzen System by LadyAnne
Mar 30
new question: A problem about Euler's totient function by mbhatia
Comments
The graph
Hi all adept graphists, please make a graph in the entry "example of jump discontinuity"! Also in "function x^x", it were nice to see the graph of the function.
Regards,
Jussi
Two new plots
Stevecheng has friendly made two fine graphs, in "function x^x" and "example of jump discontinuity". Especially the latter is superb, showing the unconventional behaviour of the function near the origin.
Thank you very much, Steve!
Jussi
Re: Two new plots
You're welcome. If anybody else would like some graphs for their entries, please feel free to ask also. (e.g. I think the entries on Riemann/Lebesgue integral / Riemann sum could seriously use some illustrations.)
// Steve
Re: Two new plots
I've added you to my editor group. That gives you write access to all my entries  feel free to add graphs wherever you think they would be appropriate.