Length measurement: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>E-Soter
m linking Oussama Khatib
en>Tom.Reding
m WP:GenFixes (page/s, endash, &nbsp, unicodify, and/or concising wikilinks, etc.), et al., ed., ref cleanup using AWB
 
Line 1: Line 1:
In [[non-standard analysis]], a discipline within [[classical mathematics]], '''microcontinuity''' (or ''S''-continuity) of an [[internal function]] ''f'' at a point ''a'' is defined as follows:
Valentine's Day is from the corner again and now you may rack your brains to need to what associated with gift choose from for your loved one girlfriend. Truly it in reality is an uphill task prefer an ideal gift since most ladies are picky and hard to please.<br><br>ugg news Fourthly, you are able to smell a pair of boots. I do advise of which you take a whiff anyone put built in. Besides, you must not confuse the smell of your personal personal foot the particular odor from the chemicals once had color synthetic fabrics. The scent resembles that of paint and the original boots are obviously odorless.<br><br>There are frequently ways in order to locate affordable ugg boots. Visit reputable websites that sell these sneakers at inexpensive prices. Generally speaking, these sites will inform you whether their boots are genuine or fake. The explanation these sites provide items at cheap prices simply because usually think of yourself as an affiliate for vehicle that manufacturers boots.<br><br>In fact, in order to find cheap boots, you should also visit those outlet sites. Outlet store usually have plenty overstocked items also in order to promote sales volume, they can possibly sell their products at more affordable prices. Compared to those products prices you normally find at retail stores, prices at outlet stores are cheaper and more. So if you are planning to buy a associated with such shoes, you can check out outlet stores in your areas to start with. If you can really find such boots at a discounted price or at a discounted price, it really can be very exciting.<br><br>Boots apparel is No more struggling to acquire it. The right associated with fashion boots apparel can be just what you require to complete your apparel that has. They look stylish whether these kinds of balancing using a sarong or peeking out of under jeans. Before you spend your hard earned money, you've got notice down some points to consider for making the importance investment on the pair of fashion boots apparel.<br><br>So don'y worried understands good bye to your ugg boots, the UGGs discount boots will acompany you for that whole season and healthiness is the main year, I think you could possibly have the same feelings.<br><br>In case you cherished this post and you wish to receive more details about [http://horizonafrica.com/img/ http://horizonafrica.com] kindly stop by our web-site.
:for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is infinitely close to ''f''(''a'').
Here ''x'' runs through the domain of ''f''.
 
For a function ''f'' defined on <math>\mathbb{R}</math>, the definition can be expressed in terms of the [[Halo (mathematics)|halo]] as follows: ''f'' is microcontinuous at <math>c\in\mathbb{R}</math> if and only if <math>f(hal(c))\subset hal(f(c))</math>.
 
==History==
 
The modern property of continuity of a function was first defined by Bolzano in 1817. However, Bolzano's work was not noticed by the larger mathematical community until its rediscovery in Heine in the 1860s. Meanwhile, [[Cauchy]]'s textbook [[Cours d'Analyse]] defined continuity in 1821 using [[infinitesimal]]s as above.<ref>{{citation
| last1 = Borovik | first1 = Alexandre
| author1-link = Alexandre Borovik
| last2 = Katz | first2 = Mikhail G.
| author2-link = Mikhail Katz
| arxiv = 1108.2885
| doi = 10.1007/s10699-011-9235-x
| issue =
| journal = [[Foundations of Science]]
| pages =
| title = Who gave you the Cauchy--Weierstrass tale? The dual history of rigorous calculus
| volume =
| year = 2011}}.</ref>
 
==Continuity and uniform continuity==
 
The property of microcontinuity is typically applied to the natural extension ''f*'' of a real function ''f''. Thus, ''f'' defined on a real interval ''I'' is continuous if and only if ''f*''  is microcontinuous at every point of ''I''. Meanwhile, ''f'' is [[uniformly continuous]] on ''I'' if and only if ''f*'' is microcontinuous at every point (standard and non-standard) of the natural extension ''I*'' of its domain ''I'' (see Davis, 1977, p.&nbsp;96).
 
==Example 1==
The real function <math>f(x)=\frac{1}{x}</math> on the open interval (0,1) is not uniformly continuous because the natural extension ''f*'' of ''f'' fails to be microcontinuous at an [[infinitesimal]] <math>a>0</math>. Indeed, for such an ''a'', the values ''a'' and ''2a'' are infinitely close, but the values of ''f*'', namely <math>\frac{1}{a}</math> and <math>\frac{1}{2a}</math> are not infinitely close.
 
==Example 2==
The function <math>f(x)=x^2</math> on <math>\mathbb{R}</math> is not uniformly continuous because ''f*'' fails to be microcontinuous at an infinite point <math>H\in \mathbb{R}^*</math>.  Namely, setting <math>e=\frac{1}{H}</math> and ''K''&nbsp;=&nbsp;''H''&nbsp;+&nbsp;''e'', one easily sees that ''H'' and ''K'' are infinitely close but ''f''*(''H'') and ''f''*(''K'') are not infinitely close.
 
==Uniform convergence==
[[Uniform convergence]] similarly admits a simplified definition in a hyperreal setting. Thus, a sequence <math>f_n</math> converges to ''f'' uniformly if for all ''x'' in the domain of ''f*'' and all infinite ''n'', <math>f_n^*(x)</math> is infinitely close to <math>f^*(x)</math>.
 
==Bibliography==
*[[Martin Davis]] (1977) Applied nonstandard analysis. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney. xii+181 pp. ISBN 0-471-19897-8
 
* Gordon, E. I.; Kusraev, A. G.; [[Semen Samsonovich Kutateladze|Kutateladze]], S. S.: Infinitesimal analysis. Updated and revised translation of the 2001 Russian original. Translated by Kutateladze. Mathematics and its Applications, 544. Kluwer Academic Publishers, Dordrecht, 2002.
 
== References ==
{{reflist}}
 
{{Infinitesimals}}
 
[[Category:Non-standard analysis]]

Latest revision as of 20:03, 7 January 2015

Valentine's Day is from the corner again and now you may rack your brains to need to what associated with gift choose from for your loved one girlfriend. Truly it in reality is an uphill task prefer an ideal gift since most ladies are picky and hard to please.

ugg news Fourthly, you are able to smell a pair of boots. I do advise of which you take a whiff anyone put built in. Besides, you must not confuse the smell of your personal personal foot the particular odor from the chemicals once had color synthetic fabrics. The scent resembles that of paint and the original boots are obviously odorless.

There are frequently ways in order to locate affordable ugg boots. Visit reputable websites that sell these sneakers at inexpensive prices. Generally speaking, these sites will inform you whether their boots are genuine or fake. The explanation these sites provide items at cheap prices simply because usually think of yourself as an affiliate for vehicle that manufacturers boots.

In fact, in order to find cheap boots, you should also visit those outlet sites. Outlet store usually have plenty overstocked items also in order to promote sales volume, they can possibly sell their products at more affordable prices. Compared to those products prices you normally find at retail stores, prices at outlet stores are cheaper and more. So if you are planning to buy a associated with such shoes, you can check out outlet stores in your areas to start with. If you can really find such boots at a discounted price or at a discounted price, it really can be very exciting.

Boots apparel is No more struggling to acquire it. The right associated with fashion boots apparel can be just what you require to complete your apparel that has. They look stylish whether these kinds of balancing using a sarong or peeking out of under jeans. Before you spend your hard earned money, you've got notice down some points to consider for making the importance investment on the pair of fashion boots apparel.

So don'y worried understands good bye to your ugg boots, the UGGs discount boots will acompany you for that whole season and healthiness is the main year, I think you could possibly have the same feelings.

In case you cherished this post and you wish to receive more details about http://horizonafrica.com kindly stop by our web-site.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Length_measurement&oldid=269467"