Strict differentiability

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".

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". 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) (i.e. continuously differentiable). In analogy with the Fréchet derivative, the previous definition can be generalized to the case where R is replaced by a Banach space E (such as \mathbb{R}^n ), and requiring existence of a continuous linear map L such that f(x)-f(y)=L(x-y)+\operatorname{o}\limits_{(x,y)\to(a,a)}(|x-y|) where o(\cdot) is defined in a natural way on E × E. F(x) = \begin{cases} p^2 & \text{if } x \equiv p \pmod{p^3} \\ p^4 & \text{if } x \equiv p^2 \pmod{p^5} \\ p^6 & \text{if } x \equiv p^3 \pmod{p^7} \\ \vdots & \vdots \\ 0 & \text{otherwise}.\end{cases} One checks that the derivative of F, according to the 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 = p n − p 2n and y = p n , 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. 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 \lim_{(x,y) \to (a,a)} \frac{F(y)-F(x)}{y-x} exists.

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