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[parent] example of jump discontinuity (Example)

The elementary real function

$\displaystyle f\colon x \mapsto \frac{1}{1+e^\frac{1}{x}}$
has a jump discontinuity at the origin, since
$\displaystyle \lim_{x\to 0-}f(x) = 1\quad \mathrm{and}\quad \lim_{x\to 0+}f(x) =0.$
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
$\displaystyle 1:(1+e^{\frac{1}{x}}) = \sum_{k=0}^\infty(-1)^ke^{\frac{k}{x}} = 1-e^{\frac{1}{x}}+e^{\frac{2}{x}}-e^{\frac{3}{x}}+-\ldots$    

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

$\displaystyle f'(x) = \frac{1}{x^2(e^{-\frac{1}{x}}+1)(1+e^\frac{1}{x})},$
and thus we have the one-sided limits $ \displaystyle \lim_{x\to 0\pm}f'(x) = 0$ (see growth of exponential function).
Figure: Graph of the function $ f$ with jump discontinuity
\includegraphics{e1xjump.eps}



"example of jump discontinuity" is owned by pahio. [ full author list (3) ]
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See Also: exponential function, improper limits


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Cross-references: growth of exponential function, one-sided limits, derivative, divisions, function, series, origin, jump discontinuity, real function
There is 1 reference to this entry.

This is version 13 of example of jump discontinuity, born on 2006-11-18, modified 2007-09-05.
Object id is 8567, canonical name is ExampleOfJumpDiscontinuity.
Accessed 3511 times total.

Classification:
AMS MSC54C05 (General topology :: Maps and general types of spaces defined by maps :: Continuous maps)
 26A15 (Real functions :: Functions of one variable :: Continuity and related questions )

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Two new plots by pahio on 2006-12-15 04:01:58
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
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The graph by pahio on 2006-11-18 11:12:43
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
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