We consider the function
-
given by
-
![{\displaystyle {}f(t):={\begin{cases}0{\text{ for }}t=0,\\{\frac {1}{t}}\sin {\frac {1}{t^{2}}}{\text{ for }}t\neq 0\,.\end{cases}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86867e2de2a7e2813c5c4193ae24e10c77cf6083)
This function is not
Riemann-integrable,
because it it neither bounded from above nor from below. Hence, there exist no
upper step functions
for
. However,
still has a
primitive function.
To see this, we consider the function
-
![{\displaystyle {}H(t):={\begin{cases}0{\text{ for }}t=0,\\{\frac {t^{2}}{2}}\cos {\frac {1}{t^{2}}}{\text{ for }}t\neq 0\,.\end{cases}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94656451c7cab6e97c8c74701585eafd99449308)
This function is
differentiable.
For
,
the derivative is
-
![{\displaystyle {}H'(t)=t\cos {\frac {1}{t^{2}}}+{\frac {1}{t}}\sin {\frac {1}{t^{2}}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/389a296c90d8bb377b001e76c80dcd0d1fe8e7b3)
For
,
the
difference quotient
is
-
![{\displaystyle {}{\frac {{\frac {h^{2}}{2}}\cos {\frac {1}{h^{2}}}}{h}}={\frac {h}{2}}\cos {\frac {1}{h^{2}}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d88888e5fc43d1d276b94a1aa9f4ec5e7ecc428)
For
, the
limit
exists and equals
, so that
is differentiable everywhere
(but not continuously differentiable).
The first summand in
is
continuous,
and therefore, due to
fact,
it has a primitive function
. Hence
is a primitive function for
. This follows for
from the explicit derivative and for
from
-
![{\displaystyle {}H'(0)-G'(0)=0-0=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f349d56c484c046b2e1e59275d05c6738bd0c1e0)