The elementary real function

has a jump discontinuity at the origin, since

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

and for  $x>0$  the series

Note.  The derivative of the function may be written as

and thus we have the one-sided limits  $\displaystyle\lim_{{x\to 0\pm}}f^{{\prime}}(x)=0$ (see growth of exponential function).