Strict differentiability

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In mathematics, strict differentiability is a modification of the usual notion of differentiability of functions that is particularly suited to p-adic analysis. In short, the definition is made more restrictive by allowing both points used in the difference quotient to "move".

1 Basic definition
2 Motivation from p-adic analysis
3 Definition in p-adic case
4 References

[edit] Basic definition

The simplest setting in which strict differentiability can be considered, is that of a real-valued function defined on an interval I of the real line. The function f:I→R is said strictly differentiable in a point a∈I if

$\lim_{(x,y)\to(a,a)}\frac{f(x)-f(y)}{x-y}$

exists, where $(x,y)\to(a,a)$ is to be considered as limit in $\mathbf{R}^2$ , and of course requiring $x\ne y$ .

A strictly differentiable function is obviously differentiable, but the converse is wrong, as can be seen from the counter-example $f(x)=x^2\sin\tfrac{1}{x},\ f(0)=0,~x_n=\tfrac{1}{(n+\frac12)\pi},\ y_n=x_{n+1}$ .

One has however the equivalence of strict differentiability on an interval I, and being of differentiability class $C 1 (I)$ .

The previous definition can be generalized to the case where R is replaced by a normed vector space E, and requiring existence of a continuous linear map L such that

f (x) - f (y) = L (x - y) + o ((x, y) - (a, a))

where $o(\cdot)$ is defined in a natural way on E×E.

[edit] Motivation from p-adic analysis

In the p-adic setting, the usual definition of the derivative fails to have certain desirable properties. For instance, it is possible for a function that is not locally constant to have zero derivative everywhere. An example of this is furnished by the function F: Z_p → Z_p, where Z_p is the ring of p-adic integers, defined by

$F(x) = \begin{cases} p^2 & \mbox{if } x \equiv p \pmod{p^3} \\ p^4 & \mbox{if } x \equiv p^2 \pmod{p^5} \\ p^6 & \mbox{if } x \equiv p^3 \pmod{p^7} \\ \vdots & \vdots \\ 0 & \mbox{otherwise}.\end{cases}$

One checks that the derivative of F, according to usual definition of the derivative, exists and is zero everywhere, including at x = 0. That is, for any x in Z_p,

$\lim_{h \to 0} \frac{F(x+h) - F(x)}{h} = 0.$

Nevertheless F fails to be locally constant at the origin.

The problem with this function is that the difference quotients

$\frac{F(y)-F(x)}{y-x}$

do not approach zero for x and y close to zero. For example, taking x = pⁿ − p²ⁿ and y = pⁿ, we have

$\frac{F(y)-F(x)}{y-x} = \frac{p^{2n} - 0}{p^n-(p^n - p^{2n})} = 1,$

which does not approach zero. The definition of strict differentiability avoids this problem by imposing a condition directly on the difference quotients.

[edit] Definition in p-adic case

Let K be a complete extension of Q_p (for example K = C_p), and let X be a subset of K with no isolated points. Then a function F : X → K is said to be strictly differentiable at x = a if the limit