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".
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)
f(x)-f(y) | |
x-y |
(x,y)\to(a,a)
R2
x\ney
A strictly differentiable function is obviously differentiable, but the converse is wrong, as can be seen from the counter-example
2\sin\tfrac{1}{x}, f(0)=0,~x | ||||
f(x)=x | ||||
|
n+1.
C1(I)
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
Rn
f(x)-f(y)=L(x-y)+\operatorname{o}\limits(x,y)\to(a,a)(|x-y|)
o( ⋅ )
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: Zp → Zp, where Zp is the ring of p-adic integers, defined by
F(x)=\begin{cases} p2&ifx\equivp\pmod{p3}\ p4&ifx\equivp2\pmod{p5}\ p6&ifx\equivp3\pmod{p7}\ \vdots&\vdots\\ 0&otherwise.\end{cases}
\limh
F(x+h)-F(x) | |
h |
=0.
The problem with this function is that the difference quotients
F(y)-F(x) | |
y-x |
F(y)-F(x) | |
y-x |
=
p2n-0 | |
pn-(pn-p2n) |
=1,
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 : X → K is said to be strictly differentiable at x = a if the limit
\lim(x,y)
F(y)-F(x) | |
y-x |