**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*

and

Therefore,

*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)*

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Tags: calculus, derivative, real analysis

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