**Statement of the problem:**

is Lipchitz, then

for some constant C.

**Proof:**

We first state a** lemma** here.

,

for some .

Because we do not know whether exists.

**Proof for lemma:**

Without loss of generality, we assume .

Then we have to prove:

Set , then we have to prove:

,

which is equivalent to:

,

use the formula that, there exists a , s.t.

.

The lemma is proved.

Let’s prove the original problem. By the lemma,

,

,

The first term is zero!

Then on every interval, the difference is of order three!

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0.000000

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