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are easier to prove) and many other things. It also is quite nice for computability theory.
Theorem 2: Function is differentiable times at a point , iff differentiable functions on the segments and can be extended to .
Proof:
Whether is differentiable at a point is clear. Now obviously, it is times differentiable at some point iff as well as exhibited the same values in .
Now, it is still true for the first derivatives that and are covarying, for which only one must be nonzero. If it is nonzero at some , then it is nonzero at some point between and , hence it is nonzero at and so that is a multiple root.
Therefore is the smallest number of times for which is differentiable.
Definition 4: Function is differentiable times at a point , iff differentiable functions can be extended to , but not necessarily by the same amount.
Corollary 1: If is differentiable times at some point , then , where is the dimension of .
Proof:
One immediately takes all partial orders under which the dimension is at most , and it remains to show that one arrives at , in which is differentiable times at that point. This is clear, because for every there is a neighborhood around such that is differentiable at every point of , and this element will be transformed by these differentiable functions to .
Theorem 3: Jack’s theorem: If is differentiable times at a point of , then there is a well-ordering of the cosets of in such that is differentiable times at that point.
Proof:
Let where all ’s are different and such that , the variation of the ’s to arrive at is greater than or equal to the difference in order of the ’s, which is . It is clear that the ’s are in increasing order, and so if one chooses to be the maximum of the ordinals of the ’s, then one can write whenever and have the same , and also whenever is for a from one of the first rows in . One then can look at the relations, for all .
Thus, let where , , is a number from , and for every $
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