- Strict differentiability
In
mathematics , strict differentiability is a modification of the usual notion of differentiability of functions that is particularly suited top-adic analysis . In short, the definition is made more restrictive by allowing both points used in thedifference quotient to "move".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: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 .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:where is defined in a natural way on "E×E".
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 integer s, 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 Z"p",: 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" = "p""n" − "p"2"n" and "y" = "p""n", 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
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: exists.
References
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