Problem? Find all differentiable functions such that
for all .
Solution? First we prove two lemmas.
Lemma 1. Suppose a differentiable function satisfies for all . If (), then and .
Proof of lemma 1. For any , there exists such that for and for . So, if , we have
Since , then
Since we choose arbitrarily, then , which implies . This further implies .
Lemma 2. If (), then is constant on .
Proof of lemma 2. From lemma 1, we know that . Call a number good if and . From lemma 1, if and are good, then is good.
We prove by induction on that for any positive integer , the number is good.
This is clearly true for . Suppose it is true for smaller . Then and are good, which implies that is good, as desired.
The numbers of the form are dense on . By continuity, we get that is constant on . Hence, lemma 2 is proved.
Now we will prove that is a quadratic function. Assume the opposite. Then there exists an interval such that does not coincide with any quadratic function. Let be such that and . Let . Then . It is easy to see that satisies . By lemma 2, is constant, say . Thus for all , which is a contradiction to the assumption that does not coincide with any quadratic function.
Therefore, must be a quadratic function. Conversely, it is easy to check that any quadratic function satisfies the required properties.
Source? A. Y. Dorogvstev, Mathematical Analysis (Russian)