Strict differentiability

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

${\displaystyle \lim _{(x,y)\to (a,a)}{\frac {f(x)-f(y)}{x-y}}}$

exists, where ${\displaystyle (x,y)\to (a,a)}$ is to be considered as limit in ${\displaystyle \mathbf {R} ^{2}}$, and of course requiring ${\displaystyle x\neq y}$.

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

${\displaystyle f(x)=x^{2}\sin {\tfrac {1}{x}},\ f(0)=0,~x_{n}={\tfrac {1}{(n+{\frac {1}{2}})\pi }},\ y_{n}=x_{n+1}.}$

One has however the equivalence of strict differentiability on an interval I, and being of differentiability class ${\displaystyle C^{1}(I)}$ (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 ${\displaystyle \mathbb {R} ^{n}}$), and requiring existence of a continuous linear map L such that

${\displaystyle f(x)-f(y)=L(x-y)+\operatorname {o} \limits _{(x,y)\to (a,a)}(|x-y|)}$

where ${\displaystyle o(\cdot )}$ is defined in a natural way on E × E.

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 integers, defined by

${\displaystyle F(x)={\begin{cases}p^{2}&{\text{if }}x\equiv p{\pmod {p^{3}}}\\p^{4}&{\text{if }}x\equiv p^{2}{\pmod {p^{5}}}\\p^{6}&{\text{if }}x\equiv p^{3}{\pmod {p^{7}}}\\\vdots &\vdots \\0&{\text{otherwise}}.\end{cases}}}$

One checks that the derivative of F, according to usual definition of the derivative, exists and is zero everywhere, including at x = 0. That is, for any x in Z_p,

${\displaystyle \lim _{h\to 0}{\frac {F(x+h)-F(x)}{h}}=0.}$

Nevertheless F fails to be locally constant at the origin.

The problem with this function is that the difference quotients

${\displaystyle {\frac {F(y)-F(x)}{y-x}}}$

do not approach zero for x and y close to zero. For example, taking x = pⁿ − p²ⁿ and y = pⁿ, we have

${\displaystyle {\frac {F(y)-F(x)}{y-x}}={\frac {p^{2n}-0}{p^{n}-(p^{n}-p^{2n})}}=1,}$

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

${\displaystyle \lim _{(x,y)\to (a,a)}{\frac {F(y)-F(x)}{y-x}}}$

exists.

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