Strictly differentiable

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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.

[edit] Motivation

In the p-adic setting, the usual definition of the derivative fails have 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

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