We consider the function
-
given by
-

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
-

This function is
differentiable.
For
,
the derivative is
-

For
,
the
difference quotient
is
-

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
-
