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".

Basic definition

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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

 

exists, where   is to be considered as limit in  , and of course requiring  .

A strictly differentiable function is obviously differentiable, but the converse is wrong, as can be seen from the counter-example

 

One has however the equivalence of strict differentiability on an interval I, and being of differentiability class   (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  ), and requiring existence of a continuous linear map L such that

 

where   is defined in a natural way on E × E.

Motivation from p-adic analysis

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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: ZpZp, where Zp is the ring of p-adic integers, defined by

 

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 Zp,

 

Nevertheless F fails to be locally constant at the origin.

The problem with this function is that the difference quotients

 

do not approach zero for x and y close to zero. For example, taking x = pnp2n and y = pn, we have

 

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

Definition in p-adic case

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

 

exists.

References

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  • Alain M. Robert (2000). A Course in p-adic Analysis. Springer. ISBN 0-387-98669-3.