We consider the function
-
given by
-
![{\displaystyle {}f(x):={\begin{cases}0,\,{\text{ if }}x\leq 0\,,\\e^{-{\frac {1}{x}}},\,{\text{ if }}x>0\,.\end{cases}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6265d4e0e367d77613d5b66b5216ac779e4b14e)
We claim that this function is infinitely often
differentiable,
which is only in
not directly clear. We first show, by induction, that all derivatives of
have the form
with certain polynomials
,
and that therefore the
limit
for
equals
(see
exercise
and
exercise).
Therefore, the limit exists for all derivatives and is
. So all derivatives in
have value
, and therefore the
Taylor series
in
is just the
zero series.
However, the Function
is in no neighborhood of
the zero function, since
.