66.59.127.126: /* See also */

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See also

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	In [[mathematics]], '''strict differentiability''' is a modification of the usual notion of [[differentiable function\|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 author is known as Wilber Pegues. I've usually loved residing in Kentucky but now I'm considering other choices. What me and my family adore is bungee jumping but I've been using on new things lately. Credit authorising is how she makes a living.<br><br>Also visit my site; psychic phone ([http://www.familysurvivalgroup.com/easy-methods-planting-looking-backyard/ familysurvivalgroup.com])

	~~== 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~~
	~~:<math>\lim_{(x,y)\to(a,a)}\frac{f(x)-f(y)}{x-y}</math>~~
	~~exists, where <math>(x,y)\to(a,a)</math> is to be considered as limit in <math>\mathbf{R}^2</math>,~~ and ~~of course requiring <math>x\ne y</math>.~~

	~~A strictly differentiable function~~ is ~~obviously differentiable,~~ but ~~the converse is wrong, as can be seen from the counter-example <math>f(x)=x^2\sin\tfrac{1}{x},\ f(0)=0,~x_n=\tfrac{1}{(n+\frac12)\pi},\ y_n=x_{n+1}</math>.~~

	~~One has however the equivalence of strict differentiability on an interval ''~~I'~~', and being of [[differentiability class]] <math>C^1(I)</math>.~~

	~~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~~
	~~:<math>f(x)-f(y)=L(x-y)+o((x,y)-(a,a))</math>~~
	~~where <math>o(\cdot)</math> 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'''~~<~~sub~~>~~''p''~~<~~/sub~~> ~~&rarr~~; ~~'''Z'''<sub>''p''</sub>, where '''Z'''<sub>''p''</sub> is the ring of [~~[~~p-adic integer]]s, defined by~~
	: ~~<math>F(x) = \begin{cases}~~
	~~p^2 & \mbox{if } x \equiv p \pmod{p^3} \\~~
	~~p^4 & \mbox{if } x \equiv p^2 \pmod{p^5} \\~~
	~~p^6 & \mbox{if } x \equiv p^3 \pmod{p^7} \\~~
	~~\vdots & \vdots \\~~
	~~0 & \mbox{otherwise}.\end{cases} <~~/~~math>~~
	~~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'''<sub>''p''<~~/~~sub>,~~
	~~: <math> \lim_{h \to 0} \frac{F(x+h) - F(x)}{h} = 0~~.~~</math>~~
	~~Nevertheless ''F'' ''fails to be locally constant'' at the origin~~.

	~~The problem with this function is that the ''difference quotients''~~
	~~: <math>\frac{F(y)-F(x)}{y-x}</math>~~
	~~do not approach zero for ''x'' and ''y'' close to zero. For example, taking ''x'' = ''p''<sup>''n''</sup> − ''p''<sup>2''n''<~~/~~sup> and ''y'' = ''p''<sup>''n''</sup>, we have~~
	~~: <math>\frac{F(y)~~-~~F(x)}{y~~-~~x} = \frac{p^{2n}~~ - ~~0}{p^n~~-~~(p^n - p^{2n})} = 1,<~~/~~math>~~
	~~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'''<sub>''p''</sub> (for example ''K'' = '''C'''<sub>''p''</sub>), 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
	~~: <math>\lim_{(x,y~~) ~~\to (a,a)} \frac{F(y)-F(x)}{y-x}</math>~~
	~~exists.~~

	~~== References ==~~
	* {{cite book \| author=Alain M. Robert \| title= A Course in ''p''-adic Analysis \| publisher= Springer \| year= 2000 \| isbn= 0-387-98669-3 }}

	~~[[Category:Number theory]]~~

en>Hans Adler: refine stub

2010-04-26T13:33:42Z

refine stub

New page

In [[mathematics]], '''strict differentiability''' is a modification of the usual notion of [[differentiable function|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 ==
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
:<math>\lim_{(x,y)\to(a,a)}\frac{f(x)-f(y)}{x-y}</math>
exists, where <math>(x,y)\to(a,a)</math> is to be considered as limit in <math>\mathbf{R}^2</math>, and of course requiring <math>x\ne y</math>.

A strictly differentiable function is obviously differentiable, but the converse is wrong, as can be seen from the counter-example <math>f(x)=x^2\sin\tfrac{1}{x},\ f(0)=0,~x_n=\tfrac{1}{(n+\frac12)\pi},\ y_n=x_{n+1}</math>.

One has however the equivalence of strict differentiability on an interval ''I'', and being of [[differentiability class]] <math>C^1(I)</math>.

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
:<math>f(x)-f(y)=L(x-y)+o((x,y)-(a,a))</math>
where <math>o(\cdot)</math> 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
: <math>F(x) = \begin{cases}
p^2 & \mbox{if } x \equiv p \pmod{p^3} \\
p^4 & \mbox{if } x \equiv p^2 \pmod{p^5} \\
p^6 & \mbox{if } x \equiv p^3 \pmod{p^7} \\
\vdots & \vdots \\
0 & \mbox{otherwise}.\end{cases} </math>
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'',
: <math> \lim_{h \to 0} \frac{F(x+h) - F(x)}{h} = 0.</math>
Nevertheless ''F'' ''fails to be locally constant'' at the origin.

The problem with this function is that the ''difference quotients''
: <math>\frac{F(y)-F(x)}{y-x}</math>
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
: <math>\frac{F(y)-F(x)}{y-x} = \frac{p^{2n} - 0}{p^n-(p^n - p^{2n})} = 1,</math>
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
: <math>\lim_{(x,y) \to (a,a)} \frac{F(y)-F(x)}{y-x}</math>
exists.

== References ==
* {{cite book | author=Alain M. Robert | title= A Course in ''p''-adic Analysis | publisher= Springer | year= 2000 | isbn= 0-387-98669-3 }}

[[Category:Number theory]]

Continuum function - Revision history

66.59.127.126: /* See also */

en>Hans Adler: refine stub