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teorema di cauchy

are easier to prove) and many other things. It also is quite nice for computability theory.

Theorem 2: Function $f$ is differentiable $k$ times at a point $\tau$, iff $k$ differentiable functions on the segments $f(0)$ and $f(1)$ can be extended to $f(\tau)$.

Proof:

Whether $f$ is differentiable at a point $\tau$ is clear. Now obviously, it is $k$ times differentiable at some point $a\in [a_i,a_{i+1}]$ iff $\frac{da(f)}{d\tau}$ as well as $\frac{da(f)}{d\tau/(a_{i+1},\tau)}$ exhibited the same values in $[a_{i+1}]$.

Now, it is still true for the first $k$ derivatives $d(f)$ that $d(f)/d\tau$ and $d(f)/d\tau/(a_{i+1},\tau)$ are covarying, for which only one must be nonzero. If it is nonzero at some $a\in [F,G]$, then it is nonzero at some point between $G$ and $\tau$, hence it is nonzero at $\tau$ and so that $\tau$ is a multiple root.

Therefore $k$ is the smallest number of times for which $f$ is differentiable. $\blacksquare$

Definition 4: Function $f$ is differentiable $k$ times at a point $\tau$, iff $k$ differentiable functions can be extended to $f(\tau)$, but not necessarily by the same amount.

Corollary 1: If $f:\mathbb{A}\to\mathbb{A}$ is differentiable $k$ times at some point $\tau$, then $k=n$, where $n$ is the dimension of $\mathbb{A}$.

Proof:

One immediately takes all partial orders under which the dimension is at most $2$, and it remains to show that one arrives at $B_{k}$, in which $f$ is differentiable $k$ times at that point. This is clear, because for every $x\in [B_{k}]$ there is a neighborhood $U_{k}$ around $\tau$ such that $f$ is differentiable at every point of $U_{k}$, and this element will be transformed by these differentiable functions to $x$.

$\blacksquare$

Theorem 3: Jack’s theorem: If $f:A\to B$ is differentiable $k$ times at a point of $A\cap B$, then there is a well-ordering of the cosets of $A$ in $B$ such that $f$ is differentiable $k$ times at that point.

Proof:

Let $\tau=\tau_1|\tau_2|\cdots|\tau_n$ where all $\tau_j$’s are different and such that $j=k$, the variation of the $\tau_j$’s to arrive at $\tau$ is greater than or equal to the difference in order of the ${\alpha}$’s, which is $2^{\lfloor\log |B|\rfloor}$. It is clear that the $\tau_j$’s are in increasing order, and so if one chooses $c_{i}$ to be the maximum of the ordinals of the $\tau_j$’s, then one can write $c_{i}\leq c_{j}$ whenever $c_{i}$ and $c_{j}$ have the same $i$, and also $c_{i+k}\leq c_{i}$ whenever $i$ is for a $\tau_j$ from one of the first $k$ rows in $c$. One then can look at the relations, $\tau_j\leq \tau_{j+1}$ for $\leq$ all $\tau_{j}$.

Thus, let $\tau=\tau_1|\tau_2|\cdots|\tau_k$ where $k=m$, $i\leq k$, $j$ is a number from $1,\cdots,m$, and for every $

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