Power MOSFET - Revision history

en>KylieTastic: Reverted edits by 130.88.114.243 (talk): editing tests (HG)

2014-12-31T16:36:26Z

Reverted edits by 130.88.114.243 (talk): editing tests (HG)

← Older revision		Revision as of 18:36, 31 December 2014
Line 1:		Line 1:
	~~Hello and welcome. My name~~ is Figures ~~Wunder. Her family members lives in Minnesota~~. ~~Playing baseball~~ is the ~~pastime he will never quit performing~~. ~~For years he's~~ been ~~operating as~~ a ~~receptionist~~.<br><br>~~Feel free to visit~~ my ~~website~~ [http://www.~~hooddirectory~~.com/~~how-you-can-cure-an-unpleasant-yeast-infection~~/ ~~hooddirectory~~.~~com~~]		The title of the writer is Figures. To gather cash is one of the things I adore most. Managing people has been his working day occupation for a whilst. South Dakota is where me and my spouse live.<br><br>Visit my web site: at home std testing ([http://www.eyelevelfederalway.com/xe/index.php?mid=Promotions&page=1&document_srl=571267 Click At this website])

en>Do ut des: /* Gate to drain capacitance */ Linking directly to subsection Parallel-plate model of Capacitor makes more sense then to the Article

2014-02-27T20:17:52Z

Gate to drain capacitance: Linking directly to subsection Parallel-plate model of Capacitor makes more sense then to the Article

← Older revision		Revision as of 22:17, 27 February 2014
Line 1:		Line 1:
	~~{{redirect\|Jump point\|the science-fiction concept\|jump drive}}~~		Hello and welcome. My name is Figures Wunder. Her family members lives in Minnesota. Playing baseball is the pastime he will never quit performing. For years he's been operating as a receptionist.<br><br>Feel free to visit my website [http://www.hooddirectory.com/how-you-can-cure-an-unpleasant-yeast-infection/ hooddirectory.com]
	~~{{Refimprove\|date=March 2013}}~~

	~~[[Continuous function]]s are of utmost importance in [[mathematics]], functions~~ and ~~applications. However, not all [[function (mathematics)\|functions]] are continuous~~. ~~If a function~~ is not continuous at a point in its [[domain (mathematics)\|domain]], one says that it has a '''discontinuity''' there. The set of all points of discontinuity of a function may be a [[discrete set]], a [[dense set]], or even the entire domain of the function.

	~~This article describes the '''classification of discontinuities'''~~ in ~~the simplest case of functions of a single [[real number\|real]] variable taking real values~~.

	~~Consider a real valued function ''ƒ'' of a real variable ''x'', defined in a neighborhood of the point ''x''<sub>0</sub> at which ''ƒ''~~ is ~~discontinuous. Three situations can be distinguished:~~

	~~{{ordered list~~
	~~\|1= The [[one-sided limit]] from the negative direction~~
	~~:<math>L^{-}=\lim_{x\to x_0^{-}} f(x)</math>~~

	~~and the one-sided limit from~~ the ~~positive direction~~
	~~:<math>L^{+}=\lim_{x\to x_0^{+}} f(x)</math>~~

	~~at <math>\scriptstyle x_0</math> exist, are finite, and are equal to <math>\scriptstyle L \;=\; L^{-} \;=\; L^{+}</math>~~. ~~Then, if~~ '~~'ƒ''(''x''<sub>0</sub>) is not equal to <math>\scriptstyle L</math>, ''x''<sub>0</sub> is called~~ a ~~''removable discontinuity''~~. ~~This discontinuity can be 'removed to make ''ƒ'' continuous at ''x''~~<~~sub~~>0<~~/sub~~>~~', or more precisely, the function~~
	~~:<math>g(x) = \begin{cases}f(x) & x\ne x_0 \\ L & x = x_0\end{cases}</math>~~

	~~is continuous at ''x''=''x''<sub>0</sub>.~~

	\|2= The limits <math>\scriptstyle L^{-}</math> and <math>\scriptstyle L^{+}</math> exist and are finite, but not equal. Then, ''x''<sub>0</sub> is called a ''jump discontinuity'' or ''step discontinuity''. For this type of discontinuity, the function ''ƒ'' may have any value at ''x''<sub>0</sub>.

	\|3= One or both of the limits <math>\scriptstyle L^{-}</math> and <math>\scriptstyle L^{+}</math> does not exist or is infinite. Then, ''x''<sub>0</sub> is called an ''essential discontinuity'', or ''infinite discontinuity''. (This is distinct from the term ''[[essential singularity]]'' which is often used when studying [[complex analysis\|functions of complex variables]].)
	}}

	The term ''removable discontinuity'' is sometimes used by [[abuse of terminology]] for cases in which the limits in both directions exist and are equal, while the function is [[defined and undefined\|undefined]] at the point <math>\scriptstyle x_0</math>.<ref>See, for example, the last sentence in the definition given at Mathwords.[http://www.~~mathwords~~.com/r/removable_discontinuity.htm]</ref> This use is abusive because [[Continuous function\|continuity]] and discontinuity of a function are concepts defined only for points in the function's domain. Such a point not in the domain is properly named a [[removable singularity]].

	~~The [[Oscillation (mathematics)\|oscillation]] of a function at a point quantifies these discontinuities as follows:~~
	* in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
	* in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits from the two sides);
	* in an essential discontinuity, oscillation measures the failure of a limit to exist.

	~~==Examples==~~
	~~[[File:Discontinuity removable.eps.png\|thumb\|left\|The function in example 1, a removable discontinuity]]~~
	~~1. Consider the function~~
	~~:<math>f(x) = \begin{cases}~~
	~~x^2 & \mbox{ for } x < 1 \\~~
	~~0 & \mbox{ for } x = 1 \\~~
	~~2-x & \mbox{ for } x > 1~~
	~~\end{cases}</math>~~

	~~Then, the point <math>\scriptstyle x_0 \;=\; 1</math> is a ''removable discontinuity''.~~
	~~{{clear}}~~

	~~[[File:Discontinuity jump.eps.png\|thumb\|left\|The function in example 2, a jump discontinuity]]~~
	~~2. Consider the function~~
	~~:<math>f(x) = \begin{cases}~~
	~~x^2 & \mbox{ for } x < 1 \\~~
	~~0 & \mbox{ for } x = 1 \\~~
	2 - (x-~~1)^2 & \mbox{ for } x > 1~~
	~~\end{cases}</math>~~

	~~Then, the point <math>\scriptstyle x_0 \;=\; 1</math> is a ''jump discontinuity''.~~
	~~{{clear}}~~

	~~[[File:Discontinuity essential.eps.png\|thumb\|left\|The function in example 3, an essential discontinuity]]~~
	~~3. Consider the function~~
	~~:<math>f(x) = \begin{cases}~~
	~~\sin\frac{5}{x~~-~~1} & \mbox{ for } x < 1 \\~~
	~~0 & \mbox{ for } x = 1 \\~~
	~~\frac{1}{x~~-~~1} & \mbox{ for } x > 1~~
	~~\end{cases}</math>~~

	~~Then, the point <math>\scriptstyle x_0 \;=\; 1</math> is an ''essential discontinuity (sometimes called infinite discontinuity)''. For it to be~~ an ~~essential discontinuity, it would have sufficed that only one of the two one~~-~~sided limits did not exist or were infinite. However, given this example the discontinuity is also an ''essential discontinuity'' for the extension of the function into complex variables.~~
	~~{{clear}}~~

	~~==The set of discontinuities of a function==~~

	~~The set of points at which a function is continuous is always a [[G~~-~~delta set\|G<sub>δ</sub> set]]. The set of discontinuities is an [[F~~-~~sigma set\|F<sub>σ<~~/~~sub> set]].~~

	~~The set of discontinuities of a [[monotonic function]] is [[countable\|at most countable]]. This is [[Froda's theorem]].~~

	~~''[[Thomae's function]]'' is discontinuous at every [[rational point]], but continuous at every irrational point.~~

	~~The [[indicator function]] of the rationals, also known as the ''[[Dirichlet function]]'', is [[nowhere continuous\|discontinuous everywhere]].~~

	~~==See also==~~
	*[[Removable singularity]]
	*[[Mathematical singularity]]
	*[[Regular_space#Extension_by_continuity\|Extension by continuity]]

	~~==Notes==~~
	~~<references />~~

	~~==References==~~
	*{{cite book
	~~\| last = Malik~~
	~~\| first = S. C.~~
	~~\| coauthors = Arora, Savita~~
	~~\| title = Mathematical analysis, 2nd ed~~
	~~\| publisher = New York: Wiley~~
	~~\| year = 1992~~
	~~\| pages =~~
	~~\| isbn = 0-470-21858-4~~
	}}

	~~==External links==~~
	* {{planetmath reference\|title=Discontinuous\|id=4447}}
	* [http://demonstrations.wolfram.com~~/Discontinuity/ "Discontinuity"] by [[Ed Pegg, Jr.]], [[The Wolfram Demonstrations Project]], 2007.~~
	* {{MathWorld \| urlname=Discontinuity \| title=Discontinuity}}
	* {{SpringerEOM\| title=Discontinuity point \| id=Discontinuity_point \| oldid=12112 \| first=L.D. \| last=Kudryavtsev }}

	~~[[Category:Mathematical analysis]~~]

en>Dicklyon: /* Latch-up */ ce

2013-12-24T08:00:33Z

Latch-up: ce

← Older revision		Revision as of 10:00, 24 December 2013
Line 1:		Line 1:
	The ~~writer~~ is called ~~Irwin~~. ~~South Dakota~~ is ~~exactly where me~~ and ~~my spouse reside~~. ~~Bookkeeping~~ is ~~my profession~~. ~~Doing ceramics~~ is ~~what my family and I appreciate~~.<br><br>~~Feel free to visit my page~~ ... [http://www.~~hotporn123~~.com/~~user~~/~~RPocock~~ http://~~www~~.~~hotporn123~~.com/]		{{redirect\|Jump point\|the science-fiction concept\|jump drive}}
			{{Refimprove\|date=March 2013}}

			[[Continuous function]]s are of utmost importance in [[mathematics]], functions and applications. However, not all [[function (mathematics)\|functions]] are continuous. If a function is not continuous at a point in its [[domain (mathematics)\|domain]], one says that it has a '''discontinuity''' there. The set of all points of discontinuity of a function may be a [[discrete set]], a [[dense set]], or even the entire domain of the function.

			This article describes the '''classification of discontinuities''' in the simplest case of functions of a single [[real number\|real]] variable taking real values.

			Consider a real valued function ''ƒ'' of a real variable ''x'', defined in a neighborhood of the point ''x''<sub>0</sub> at which ''ƒ'' is discontinuous. Three situations can be distinguished:

			{{ordered list
			\|1= The [[one-sided limit]] from the negative direction
			:<math>L^{-}=\lim_{x\to x_0^{-}} f(x)</math>

			and the one-sided limit from the positive direction
			:<math>L^{+}=\lim_{x\to x_0^{+}} f(x)</math>

			at <math>\scriptstyle x_0</math> exist, are finite, and are equal to <math>\scriptstyle L \;=\; L^{-} \;=\; L^{+}</math>. Then, if ''ƒ''(''x''<sub>0</sub>) is not equal to <math>\scriptstyle L</math>, ''x''<sub>0</sub> is called a ''removable discontinuity''. This discontinuity can be 'removed to make ''ƒ'' continuous at ''x''<sub>0</sub>', or more precisely, the function
			:<math>g(x) = \begin{cases}f(x) & x\ne x_0 \\ L & x = x_0\end{cases}</math>

			is continuous at ''x''=''x''<sub>0</sub>.

			\|2= The limits <math>\scriptstyle L^{-}</math> and <math>\scriptstyle L^{+}</math> exist and are finite, but not equal. Then, ''x''<sub>0</sub> is called a ''jump discontinuity'' or ''step discontinuity''. For this type of discontinuity, the function ''ƒ'' may have any value at ''x''<sub>0</sub>.

			\|3= One or both of the limits <math>\scriptstyle L^{-}</math> and <math>\scriptstyle L^{+}</math> does not exist or is infinite. Then, ''x''<sub>0</sub> is called an ''essential discontinuity'', or ''infinite discontinuity''. (This is distinct from the term ''[[essential singularity]]'' which is often used when studying [[complex analysis\|functions of complex variables]].)
			}}

			The term ''removable discontinuity'' is sometimes used by [[abuse of terminology]] for cases in which the limits in both directions exist and are equal, while the function is [[defined and undefined\|undefined]] at the point <math>\scriptstyle x_0</math>.<ref>See, for example, the last sentence in the definition given at Mathwords.[http://www.mathwords.com/r/removable_discontinuity.htm]</ref> This use is abusive because [[Continuous function\|continuity]] and discontinuity of a function are concepts defined only for points in the function's domain. Such a point not in the domain is properly named a [[removable singularity]].

			The [[Oscillation (mathematics)\|oscillation]] of a function at a point quantifies these discontinuities as follows:
			* in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
			* in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits from the two sides);
			* in an essential discontinuity, oscillation measures the failure of a limit to exist.

			==Examples==
			[[File:Discontinuity removable.eps.png\|thumb\|left\|The function in example 1, a removable discontinuity]]
			1. Consider the function
			:<math>f(x) = \begin{cases}
			x^2 & \mbox{ for } x < 1 \\
			0 & \mbox{ for } x = 1 \\
			2-x & \mbox{ for } x > 1
			\end{cases}</math>

			Then, the point <math>\scriptstyle x_0 \;=\; 1</math> is a ''removable discontinuity''.
			{{clear}}

			[[File:Discontinuity jump.eps.png\|thumb\|left\|The function in example 2, a jump discontinuity]]
			2. Consider the function
			:<math>f(x) = \begin{cases}
			x^2 & \mbox{ for } x < 1 \\
			0 & \mbox{ for } x = 1 \\
			2 - (x-1)^2 & \mbox{ for } x > 1
			\end{cases}</math>

			Then, the point <math>\scriptstyle x_0 \;=\; 1</math> is a ''jump discontinuity''.
			{{clear}}

			[[File:Discontinuity essential.eps.png\|thumb\|left\|The function in example 3, an essential discontinuity]]
			3. Consider the function
			:<math>f(x) = \begin{cases}
			\sin\frac{5}{x-1} & \mbox{ for } x < 1 \\
			0 & \mbox{ for } x = 1 \\
			\frac{1}{x-1} & \mbox{ for } x > 1
			\end{cases}</math>

			Then, the point <math>\scriptstyle x_0 \;=\; 1</math> is an ''essential discontinuity (sometimes called infinite discontinuity)''. For it to be an essential discontinuity, it would have sufficed that only one of the two one-sided limits did not exist or were infinite. However, given this example the discontinuity is also an ''essential discontinuity'' for the extension of the function into complex variables.
			{{clear}}

			==The set of discontinuities of a function==

			The set of points at which a function is continuous is always a [[G-delta set\|G<sub>δ</sub> set]]. The set of discontinuities is an [[F-sigma set\|F<sub>σ</sub> set]].

			The set of discontinuities of a [[monotonic function]] is [[countable\|at most countable]]. This is [[Froda's theorem]].

			''[[Thomae's function]]'' is discontinuous at every [[rational point]], but continuous at every irrational point.

			The [[indicator function]] of the rationals, also known as the ''[[Dirichlet function]]'', is [[nowhere continuous\|discontinuous everywhere]].

			==See also==
			*[[Removable singularity]]
			*[[Mathematical singularity]]
			*[[Regular_space#Extension_by_continuity\|Extension by continuity]]

			==Notes==
			<references />

			==References==
			*{{cite book
			\| last = Malik
			\| first = S. C.
			\| coauthors = Arora, Savita
			\| title = Mathematical analysis, 2nd ed
			\| publisher = New York: Wiley
			\| year = 1992
			\| pages =
			\| isbn = 0-470-21858-4
			}}

			==External links==
			* {{planetmath reference\|title=Discontinuous\|id=4447}}
			* [http://demonstrations.wolfram.com/Discontinuity/ "Discontinuity"] by [[Ed Pegg, Jr.]], [[The Wolfram Demonstrations Project]], 2007.
			* {{MathWorld \| urlname=Discontinuity \| title=Discontinuity}}
			* {{SpringerEOM\| title=Discontinuity point \| id=Discontinuity_point \| oldid=12112 \| first=L.D. \| last=Kudryavtsev }}

			[[Category:Mathematical analysis]]

217.10.52.10 at 09:52, 20 August 2012

2012-08-20T09:52:27Z

New page

The writer is called Irwin. South Dakota is exactly where me and my spouse reside. Bookkeeping is my profession. Doing ceramics is what my family and I appreciate.<br><br>Feel free to visit my page ... [http://www.hotporn123.com/user/RPocock http://www.hotporn123.com/]