Find continuous functions which satisfy the given conditions for all .
-
.
-
.
-
.
- No such function can exist since for
we have
- Taking derivatives of both sides of the given equation we have
- Again, taking derivatives of both sides we have
at all points
such that
. (Since
is not satisfied by the zero function
, we know there are real
such that
.) Then, integrating
Now, we can solve for
by evaluating the given identity at
,
Therefore, we have