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