Separately continuous function - Revision history

Vipul at 22:03, 24 January 2013

2013-01-24T22:03:29Z

← Older revision		Revision as of 22:03, 24 January 2013
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	We say that a function <math>f</math> of several variables is ''separately continuous'' in the variables at a point if it is separately continuous with respect to each of the variables at the point.		We say that a function <math>f</math> of several variables is ''separately continuous'' in the variables at a point if it is separately continuous with respect to each of the variables at the point.

	<center>{{#widget:YouTube\|id=~~IhLvsOcFXQ0~~}}</center>		<center>{{#widget:YouTube\|id=VK3f0Omburw}}</center>

	===For a function of two variables===		===For a function of two variables===
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	* <math>f</math> is ''separately continuous'' at the point <math>(x_0,y_0)</math> if it is continuous with respect to <math>x</math> ''and'' continuous with respect to <math>y</math> at the point <math>(x_0,y_0)</math>.		* <math>f</math> is ''separately continuous'' at the point <math>(x_0,y_0)</math> if it is continuous with respect to <math>x</math> ''and'' continuous with respect to <math>y</math> at the point <math>(x_0,y_0)</math>.

	<center>{{#widget:YouTube\|id=~~Bs3kbdjPiz0~~}}</center>		<center>{{#widget:YouTube\|id=uBTbYsHzC6Y}}</center>

	===For a function of multiple variables===		===For a function of multiple variables===
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	* We say that <math>f</math> is ''separately continuous'' in terms of ''all'' the inputs <math>x_1,x_2,\dots,x_n</math> at a point <math>(a_1,a_2,\dots,a_n)</math> if it is continuous with respect to <math>x_i</math> at <math>(a_1,a_2,\dots,a_n)</math> for each <math>i \in \{ 1,2,3,\dots,n \}</math>.		* We say that <math>f</math> is ''separately continuous'' in terms of ''all'' the inputs <math>x_1,x_2,\dots,x_n</math> at a point <math>(a_1,a_2,\dots,a_n)</math> if it is continuous with respect to <math>x_i</math> at <math>(a_1,a_2,\dots,a_n)</math> for each <math>i \in \{ 1,2,3,\dots,n \}</math>.

	<center>{{#widget:YouTube\|id=~~ABtLXnbxtuo~~}}</center>		<center>{{#widget:YouTube\|id=SJ0_tWVMmcQ}}</center>

	==Definition as a function on an open domain==		==Definition as a function on an open domain==

Vipul: /* Graphical interpretation */

2012-05-08T21:21:15Z

Graphical interpretation

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 |-
 | <math>f</math> is separately continuous in both variables everywhere. || Both the above conditions, i.e., the restrictions of the graph to all planes parallel to either the <math>xz</math>-plane or the <math>yz</math>-plane are graphs of continuous functions.
 |}

Vipul at 21:16, 8 May 2012

2012-05-08T21:16:47Z

← Older revision		Revision as of 21:16, 8 May 2012
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	* For each <math>i \in \{1,2,3,\dots,n \}</math>, we say that <math>f</math> is continuous in <math>x_i</math> if <math>f</math> is continuous in <math>x_i</math> for every point in its domain.		* For each <math>i \in \{1,2,3,\dots,n \}</math>, we say that <math>f</math> is continuous in <math>x_i</math> if <math>f</math> is continuous in <math>x_i</math> for every point in its domain.
	* We say that <math>f</math> is ''separately continuous'' in terms of ''all'' the inputs <math>x_1,x_2,\dots,x_n</math> if it is continuous in all the inputs for every point in its domain.		* We say that <math>f</math> is ''separately continuous'' in terms of ''all'' the inputs <math>x_1,x_2,\dots,x_n</math> if it is continuous in all the inputs for every point in its domain.

			==Graphical interpretation==

			===For a function of two variables===

			Suppose <math>f</math> is a function of two variables <math>x,y</math>. We consider the [[graph of a function of two variables\|graph]] of <math>f</math> as the subset <math>z = f(x,y)</math> in three-dimensional space with coordinate axes <math>x,y,z</math>.

			We have the following:

			{\| class="sortable" border="1"
			! Assertion about continuity !! How we can verify it from the graph
			\|-
			\| <math>f</math> is continuous in <math>x</math> at the point <math>(x_0,y_0)</math> \|\| Consider the graph restricted to the plane <math>y= y_0</math>. This is continuous at <math>x = x_0</math>.
			\|-
			\| <math>f</math> is continuous in <math>y</math> at the point <math>(x_0,y_0)</math> \|\| Consider the graph restricted to the plane <math>x= x_0</math>. This is continuous at <math>y = y_0</math>.
			\|-
			\| <math>f</math> is separately continuous continuous in both variables at the point <math>(x_0,y_0)</math>. \|\| Both the above conditions.
			\|-
			\| <math>f</math> is continuous in <math>x</math> everywhere. \|\| The restrictions of the graph to all planes parallel to the <math>xz</math>-plane give graphs of continuous functions.
			\|-
			\| <math>f</math> is continuous in <math>y</math> everywhere. \|\| The restrictions of the graph to all planes parallel to the <math>yz</math>-plane give graphs of continuous functions.
			\|-
			\| <math>f</math> is separately continuous in both variables everywhere. \|\| Both the above conditions, i.e., the restrictions of the graph to all planes parallel to either the <math>xz</math>-plane or the <math>yz</math>-plane are graphs of continuous functions.
			\|}

Vipul at 20:59, 14 April 2012

2012-04-14T20:59:32Z

← Older revision		Revision as of 20:59, 14 April 2012
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	We say that a function <math>f</math> of several variables is ''separately continuous'' in the variables at a point if it is separately continuous with respect to each of the variables at the point.		We say that a function <math>f</math> of several variables is ''separately continuous'' in the variables at a point if it is separately continuous with respect to each of the variables at the point.

			<center>{{#widget:YouTube\|id=IhLvsOcFXQ0}}</center>

	===For a function of two variables===		===For a function of two variables===
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	* <math>f</math> is continuous with respect to <math>y</math> at the point <math>(x_0,y_0)</math> if the function <math>y \mapsto f(x_0,y)</math> (viewed as a function of one variable <math>y</math>) is continuous at <math>y = y_0</math>.		* <math>f</math> is continuous with respect to <math>y</math> at the point <math>(x_0,y_0)</math> if the function <math>y \mapsto f(x_0,y)</math> (viewed as a function of one variable <math>y</math>) is continuous at <math>y = y_0</math>.
	* <math>f</math> is ''separately continuous'' at the point <math>(x_0,y_0)</math> if it is continuous with respect to <math>x</math> ''and'' continuous with respect to <math>y</math> at the point <math>(x_0,y_0)</math>.		* <math>f</math> is ''separately continuous'' at the point <math>(x_0,y_0)</math> if it is continuous with respect to <math>x</math> ''and'' continuous with respect to <math>y</math> at the point <math>(x_0,y_0)</math>.

			<center>{{#widget:YouTube\|id=Bs3kbdjPiz0}}</center>

	===For a function of multiple variables===		===For a function of multiple variables===
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	* For each <math>i \in \{1,2,3,\dots,n \}</math>, we say that <math>f</math> is continuous in <math>x_i</math> at the point <math>(a_1,a_2,\dots,a_n)</math> if the function <math>x_i \mapsto f(a_1,a_2,\dots,a_{i-1},x_i,a_{i+1},\dots,a_n)</math> is continuous at <math>x_i = a_i</math>.		* For each <math>i \in \{1,2,3,\dots,n \}</math>, we say that <math>f</math> is continuous in <math>x_i</math> at the point <math>(a_1,a_2,\dots,a_n)</math> if the function <math>x_i \mapsto f(a_1,a_2,\dots,a_{i-1},x_i,a_{i+1},\dots,a_n)</math> is continuous at <math>x_i = a_i</math>.
	* We say that <math>f</math> is ''separately continuous'' in terms of ''all'' the inputs <math>x_1,x_2,\dots,x_n</math> at a point <math>(a_1,a_2,\dots,a_n)</math> if it is continuous with respect to <math>x_i</math> at <math>(a_1,a_2,\dots,a_n)</math> for each <math>i \in \{ 1,2,3,\dots,n \}</math>.		* We say that <math>f</math> is ''separately continuous'' in terms of ''all'' the inputs <math>x_1,x_2,\dots,x_n</math> at a point <math>(a_1,a_2,\dots,a_n)</math> if it is continuous with respect to <math>x_i</math> at <math>(a_1,a_2,\dots,a_n)</math> for each <math>i \in \{ 1,2,3,\dots,n \}</math>.

			<center>{{#widget:YouTube\|id=ABtLXnbxtuo}}</center>

	==Definition as a function on an open domain==		==Definition as a function on an open domain==

Vipul: /* For a function of multiple variables */

2012-04-13T20:29:31Z

For a function of multiple variables

← Older revision		Revision as of 20:29, 13 April 2012
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	===For a function of multiple variables===		===For a function of multiple variables===

	Suppose <math>f</math> is a real-valued function of variables <math>x_1,x_2,\dots,x_n</math>, i.e., the [[domain]] of <math>f</matH> is a subset of <math>\R^n</math>:		Suppose <math>f</math> is a real-valued function of variables <math>x_1,x_2,\dots,x_n</math>, i.e., the [[domain]] of <math>f</matH> is an open subset of <math>\R^n</math>:

	* For each <math>i \in \{1,2,3,\dots,n \}</math>, we say that <math>f</math> is continuous in <math>x_i</math> if <math>f</math> is continuous in <math>x_i</math> for every point in its domain.		* For each <math>i \in \{1,2,3,\dots,n \}</math>, we say that <math>f</math> is continuous in <math>x_i</math> if <math>f</math> is continuous in <math>x_i</math> for every point in its domain.
	* We say that <math>f</math> is ''separately continuous'' in terms of ''all'' the inputs <math>x_1,x_2,\dots,x_n</math> if it is continuous in all the inputs for every point in its domain.		* We say that <math>f</math> is ''separately continuous'' in terms of ''all'' the inputs <math>x_1,x_2,\dots,x_n</math> if it is continuous in all the inputs for every point in its domain.

Vipul at 00:47, 13 April 2012

2012-04-13T00:47:07Z

@@ Line 17: / Line 17: @@
 ===For a function of multiple variables===
-Suppose <math>f</math> is a real-valued function of variables <math>x_1,x_2,\dots,x_n</math>. Suppose <math>(a_1,a_2,\dots,a_n)</math> is a point in the domain of <math>f</math>, i.e., it is the point where <math>x_1 = a_1, x_2 = a_2,\dots,x_n = a_n</math> (here <math>a_1,a_2,\dots,a_n</math> are actual numerical values). We define two notions:
+Suppose <math>f</math> is a real-valued function of variables <math>x_1,x_2,\dots,x_n</math>, i.e., the [[domain]] of <math>f</matH> is a subset of <math>\R^n</math>. Suppose <math>(a_1,a_2,\dots,a_n)</math> is a point in the domain of <math>f</math>, i.e., it is the point where <math>x_1 = a_1, x_2 = a_2,\dots,x_n = a_n</math> (here <math>a_1,a_2,\dots,a_n</math> are actual numerical values). We define two notions:
 * For each <math>i \in \{1,2,3,\dots,n \}</math>, we say that <math>f</math> is continuous in <math>x_i</math> at the point <math>(a_1,a_2,\dots,a_n)</math> if the function <math>x_i \mapsto f(a_1,a_2,\dots,a_{i-1},x_i,a_{i+1},\dots,a_n)</math> is continuous at <math>x_i = a_i</math>.
 * We say that <math>f</math> is ''separately continuous'' in terms of ''all'' the inputs <math>x_1,x_2,\dots,x_n</math> at a point <math>(a_1,a_2,\dots,a_n)</math> if it is continuous with respect to <math>x_i</math> at <math>(a_1,a_2,\dots,a_n)</math> for each <math>i \in \{ 1,2,3,\dots,n \}</math>.

Vipul: Created page with "==Definition at a point== ===Generic definition=== Suppose $f$ is a function of more than one variable, where $x$ is one of the input variables to
2012-04-13T00:39:50Z

Created page with "==Definition at a point== ===Generic definition=== Suppose <math>f</math> is a function of more than one variable, where <math>x</math> is one of the input variables to <mat..."

New page
==Definition at a point==

===Generic definition===

Suppose <math>f</math> is a function of more than one variable, where <math>x</math> is one of the input variables to <math>f</math>. Fix a choice <math>x = x_0</math> and fix values of all the other input variables. We say that <math>f</math> is continuous ''with respect to'' <math>x</math> at this point in its domain if the following holds: the function that sends <math>x</math> to <math>f</math> evaluated at <math>x</math> and the ''fixed'' choice of the other input variables is continuous at <math>x = x_0</math>.

We say that a function <math>f</math> of several variables is ''separately continuous'' in the variables at a point if it is separately continuous with respect to each of the variables at the point.

===For a function of two variables===

Suppose <math>f</math> is a real-valued function of two variables <math>x,y</math>, i.e., the [[domain]] of <math>f</math> is a subset of <math>\R^2</math>. Suppose <math>(x_0,y_0)</math> is a point in the domain of <math>f</math>, i.e., it is the point where <math>x = x_0</math> and <math>y = y_0</math> (here <math>x_0,y_0</math> are actual numerical values). We define three notions:

* <math>f</math> is continuous with respect to <math>x</math> at the point <math>(x_0,y_0)</math> if the function <math>x \mapsto f(x,y_0)</math> (viewed as a function of one variable <math>x</math>) is continuous at <math>x = x_0</math>.
* <math>f</math> is continuous with respect to <math>y</math> at the point <math>(x_0,y_0)</math> if the function <math>y \mapsto f(x_0,y)</math> (viewed as a function of one variable <math>y</math>) is continuous at <math>y = y_0</math>.
* <math>f</math> is ''separately continuous'' at the point <math>(x_0,y_0)</math> if it is continuous with respect to <math>x</math> ''and'' continuous with respect to <math>y</math> at the point <math>(x_0,y_0)</math>.

===For a function of multiple variables===

Suppose <math>f</math> is a real-valued function of variables <math>x_1,x_2,\dots,x_n</math>. Suppose <math>(a_1,a_2,\dots,a_n)</math> is a point in the domain of <math>f</math>, i.e., it is the point where <math>x_1 = a_1, x_2 = a_2,\dots,x_n = a_n</math> (here <math>a_1,a_2,\dots,a_n</math> are actual numerical values). We define two notions:

* For each <math>i \in \{1,2,3,\dots,n \}</math>, we say that <math>f</math> is continuous in <math>x_i</math> at the point <math>(a_1,a_2,\dots,a_n)</math> if the function <math>x_i \mapsto f(a_1,a_2,\dots,a_{i-1},x_i,a_{i+1},\dots,a_n)</math> is continuous at <math>x_i = a_i</math>.
* We say that <math>f</math> is ''separately continuous'' in terms of ''all'' the inputs <math>x_1,x_2,\dots,x_n</math> at a point <math>(a_1,a_2,\dots,a_n)</math> if it is continuous with respect to <math>x_i</math> at <math>(a_1,a_2,\dots,a_n)</math> for each <math>i \in \{ 1,2,3,\dots,n \}</math>.