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	~~In [[mathematical analysis]], a '''Pompeiu derivative''' is a real-valued [[function (mathematics)\|function]] of one real variable~~ that ~~is the [[derivative (mathematics)\|derivative]] of an everywhere [[differentiable]] function and that vanishes in a [[dense set]]~~. ~~Note in particular~~ that ~~a Pompeiu derivative is discontinuous at~~ any ~~point where it is not 0~~. ~~Whether non-identically zero such functions may exist was~~ a problem ~~that arose in~~ the ~~context~~ of ~~early-1900s research on functional differentiability~~ and ~~integrability~~. ~~The question was affirmatively answered by [[Dimitrie Pompeiu]] by constructiong an explicit example; these functions are therefore named after him~~.		There are many online companies that write custom scholarship essays for students. Term Paper Help As we have discussed before that, if you need any sort of help regarding term paper help, so you can any time contact with us and we will tackle your problems. It also helped solve a huge problem with last year's application by streamlining the mechanics of applying for colleges electing to do away with separate, and sometimes overlooked, writing supplements. Referenced material, either reworded or echoed ought to be incorporated into the paper. Therefore, you have to be careful since the first impression that you make is very important. <br><br>e articles that have been submitted several times to the site but are not 100% unique. A fine method of finishing a custom essay is with upcoming recommendations. One reason for this apprehension in the students is the fear of the 'dreaded' college essay. This doesn’t mean that spell check leaves words incorrectly spelled; it means that the word that you intended may not be the word that is included in the paper. quite well, he can become a free agent the subsequent season and create his own. <br><br>Think about how you are going to complete the task whilst simultaneously maximising the available marks in each section. com will provide a very good opportunity for college students to get involved in a greater capacity in writing their essays. provide persuasive essay tips to students in all academic levels. Students can think of taking help from their friends or some family member, to assist them in their writing assignments. Obviously it's best to write your own essays; don't let laziness goad you into taking a shortcut. <br><br>The thesis statement usually incorporates the main idea of the essay. There is customer care team that is available to the customers at any hour of the day. The main purpose of essay writing is to pass information to an identified audience in the society. If you adored this write-up and you would such as to receive even more info relating to [http://www.netcrew.co.za/index.php?mod=users&action=view&id=111575 can someone write my paper for me] kindly browse through the page. The tips on essay writing will help students write quality essays that will earn them high marks in the essay papers. This is important to ensure that you buy essay papers that are right, appropriate, and of high quality. <br><br>Be conscious of what you are actually putting down on paper. Several companies are providing companies while in the industry of cheap essay writing and guarantee lots but tend not to supply. The writers are comfortable working with any of the academic writing style you need. The nature of such visibility is clearly decided by the magnet-characteristic of its general medium of transaction. Just take a look at your essay dilemma when it is really damaged down and from right here look and feel for main sources.

	~~==Pompeiu's construction==~~
	~~Pompeiu's construction~~ is ~~described here~~. ~~Let~~ <~~math~~>~~\sqrt[3]{x}~~<~~/math~~> ~~denote~~ the ~~real [[cubic root]]~~ of the ~~real number <math>x~~.~~</math> Let <math>\{q_j\}_{j\in \N}</math>~~ be ~~an enumeration of~~ the ~~rational numbers~~ in the ~~unit interval <math>[0~~,~~\,1]~~.<~~/math~~> ~~Let~~ <~~math~~>~~\{a_j\}_{j\~~in ~~\N}</math> be positive real numbers with <math>\textstyle\sum_j a_j < \infty~~.~~</math> Define,~~ for ~~all <math>x\~~in ~~[0,\,1]</math>~~

	~~:<math>g(x):=\sum_{j=0}^\infty \,a_j \sqrt[3]{x-q_j}~~.~~</math>~~

	~~Since for any <math>x\~~in~~[0,\,1]</math> each term~~ of ~~the series is less than~~ or ~~equal~~ to ''a''<~~sub~~>j<~~/sub~~> ~~in absolute value,~~ the ~~series [[uniform convergence\|uniformly converges]] to a continuous, strictly increasing function g(x), due to~~ the ~~[[Weierstrass M-test]]~~. ~~Moreover, it turns out~~ that ~~the function ''g''~~ is ~~differentiable, with~~
	~~:<math>g^{\prime}(x):=\frac{1}{3}\sum_{j=0}^\infty \frac{a_j}{\sqrt[3]{(x-q_j)^2}}>0,</math>~~
	~~at any point where~~ the ~~sum is finite; also, at all other points, in particular,~~ at any of the ~~<math>q_j,</math> one has <math>\textstyle g^{\prime}(x):=+\infty~~.~~</math> Since the [[image (function)\|image]]~~ of ~~<math>g</math>~~ is ~~a closed bounded interval with left endpoint <math>0=g(0),</math> up~~ to ~~a multiplicative constant factor one can assume that ''g'' maps~~ the ~~interval <math>[0,\,1]</math> onto itself~~. ~~Since ''g'' is strictly increasing, it is a homeomorphism;~~ and ~~by the theorem of differentiation of the inverse function, its~~ [~~[inverse function\|composition inverse]] <math>f\,~~:~~=g^{-1}<~~/~~math> has a finite derivative at any point, which vanishes at least in the points <math>\{g(q_j)\}_{j\in \N}~~.</~~math> These form a dense subset of <math> [0,\,1]</math> (actually, it vanishes in many other points; see below)~~.

	==~~Properties~~==
	* It is known that the zero-set of a derivative of any everywhere differentiable function is a [[G delta set\|''G''<sub>δ</sub>]] ~~subset of~~ the ~~real line~~. ~~By definition, for any Pompeiu function this set is a ''dense'' G<sub>δ</sub> set, therefore by~~ the ~~[[Baire category theorem]] it~~ is ~~a [[residual set]]. In particular~~, ~~it possesses uncountably many points.~~
	* A linear combination ''af''(''x'') + ''bg''(''x'') of Pompeiu functions is a derivative, and ~~vanishes on the set {''f'' = 0} ∩ {''g'' = 0}, which is a dense ''G''~~<~~sub~~>δ<~~/sub~~> ~~by the Baire category theorem~~. ~~Thus, Pompeiu functions~~ are ~~a vector space~~ of ~~functions.~~
	* A limit function of a uniformly convergent sequence of Pompeiu derivatives is a Pompeiou derivative. Indeed, it is a derivative, due to ~~the theorem of limit under the sign of derivative~~. ~~Moreover, it vanishes in the intersection of the zero sets of the functions of the sequences: since these~~ are ~~dense ''G''<sub>δ</sub> sets, the zero set~~ of the ~~limit function is also dense~~.
	* As a consequence, the class ''E'' of ~~all bounded Pompeiu derivatives on an interval [''a'', ''b'']~~ is ~~a ''closed'' linear subspace~~ of ~~the Banach space~~ of ~~all bounded functions under the uniform distance (hence, it is a Banach space)~~.
	*Pompeiu's above construction of a ~~''positive'' function~~ is ~~a rather peculiar example of a Pompeiu's function: a theorem of Weil states that ''generically'' a Pompeiu derivative assumes both positive~~ and ~~negative values in dense sets, in the precise meaning that such functions constitute a residual set of the Banach space ''E''.~~

	~~==References==~~
	* Pompeiu, Dimitrie, "Sur les fonctions dérivées"; Math. Ann. 63 (1907), no. 3, 326—332.
	* Andrew M. Bruckner, "Differentiation of real functions"; CRM Monograph series, Montreal (1994).

	~~{{DEFAULTSORT:Pompeiu Derivative}}~~
	~~[[Category:Real analysis]]~~

en>Kiefer.Wolfowitz: /* References */

2010-04-03T16:08:55Z

References

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In [[mathematical analysis]], a '''Pompeiu derivative''' is a real-valued [[function (mathematics)|function]] of one real variable that is the [[derivative (mathematics)|derivative]] of an everywhere [[differentiable]] function and that vanishes in a [[dense set]]. Note in particular that a Pompeiu derivative is discontinuous at any point where it is not 0. Whether non-identically zero such functions may exist was a problem that arose in the context of early-1900s research on functional differentiability and integrability. The question was affirmatively answered by [[Dimitrie Pompeiu]] by constructiong an explicit example; these functions are therefore named after him.

==Pompeiu's construction==
Pompeiu's construction is described here. Let <math>\sqrt[3]{x}</math> denote the real [[cubic root]] of the real number <math>x.</math> Let <math>\{q_j\}_{j\in \N}</math> be an enumeration of the rational numbers in the unit interval <math>[0,\,1].</math> Let <math>\{a_j\}_{j\in \N}</math> be positive real numbers with <math>\textstyle\sum_j a_j < \infty.</math> Define, for all <math>x\in [0,\,1]</math>

:<math>g(x):=\sum_{j=0}^\infty \,a_j \sqrt[3]{x-q_j}.</math>

Since for any <math>x\in[0,\,1]</math> each term of the series is less than or equal to ''a''j in absolute value, the series [[uniform convergence|uniformly converges]] to a continuous, strictly increasing function g(x), due to the [[Weierstrass M-test]]. Moreover, it turns out that the function ''g'' is differentiable, with
:<math>g^{\prime}(x):=\frac{1}{3}\sum_{j=0}^\infty \frac{a_j}{\sqrt[3]{(x-q_j)^2}}>0,</math>
at any point where the sum is finite; also, at all other points, in particular, at any of the <math>q_j,</math> one has <math>\textstyle g^{\prime}(x):=+\infty.</math> Since the [[image (function)|image]] of <math>g</math> is a closed bounded interval with left endpoint <math>0=g(0),</math> up to a multiplicative constant factor one can assume that ''g'' maps the interval <math>[0,\,1]</math> onto itself. Since ''g'' is strictly increasing, it is a homeomorphism; and by the theorem of differentiation of the inverse function, its [[inverse function|composition inverse]] <math>f\,:=g^{-1}</math> has a finite derivative at any point, which vanishes at least in the points <math>\{g(q_j)\}_{j\in \N}.</math> These form a dense subset of <math> [0,\,1]</math> (actually, it vanishes in many other points; see below).

==Properties==
* It is known that the zero-set of a derivative of any everywhere differentiable function is a [[G delta set|''G''δ]] subset of the real line. By definition, for any Pompeiu function this set is a ''dense'' Gδ set, therefore by the [[Baire category theorem]] it is a [[residual set]]. In particular, it possesses uncountably many points.
* A linear combination ''af''(''x'') + ''bg''(''x'') of Pompeiu functions is a derivative, and vanishes on the set {''f'' = 0} ∩ {''g'' = 0}, which is a dense ''G''δ by the Baire category theorem. Thus, Pompeiu functions are a vector space of functions.
* A limit function of a uniformly convergent sequence of Pompeiu derivatives is a Pompeiou derivative. Indeed, it is a derivative, due to the theorem of limit under the sign of derivative. Moreover, it vanishes in the intersection of the zero sets of the functions of the sequences: since these are dense ''G''δ sets, the zero set of the limit function is also dense.
* As a consequence, the class ''E'' of all bounded Pompeiu derivatives on an interval [''a'', ''b''] is a ''closed'' linear subspace of the Banach space of all bounded functions under the uniform distance (hence, it is a Banach space).
*Pompeiu's above construction of a ''positive'' function is a rather peculiar example of a Pompeiu's function: a theorem of Weil states that ''generically'' a Pompeiu derivative assumes both positive and negative values in dense sets, in the precise meaning that such functions constitute a residual set of the Banach space ''E''.

==References==
* Pompeiu, Dimitrie, "Sur les fonctions dérivées"; Math. Ann. 63 (1907), no. 3, 326—332.
* Andrew M. Bruckner, "Differentiation of real functions"; CRM Monograph series, Montreal (1994).

{{DEFAULTSORT:Pompeiu Derivative}}
[[Category:Real analysis]]

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en>LukasMatt: Dead-end pages clean up project; you can help!

en>Kiefer.Wolfowitz: /* References */