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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).
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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.