Continuous functions form a unital algebra - Revision history

Vipul at 15:54, 16 October 2011

2011-10-16T15:54:51Z

← Older revision		Revision as of 15:54, 16 October 2011
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	* '''Additive closure''': If <math>f,g</math> are functions defined in open intervals containing <math>c</math> and both of them are [[fact about::continuous function\|continuous]] at <math>c</math>, then the [[fact about::pointwise sum of functions\|pointwise sum]] <math>f + g</math> is continuous at <math>c</math>.		* '''Additive closure''': If <math>f,g</math> are functions defined in open intervals containing <math>c</math> and both of them are [[fact about::continuous function\|continuous]] at <math>c</math>, then the [[fact about::pointwise sum of functions\|pointwise sum]] <math>f + g</math> is continuous at <math>c</math>.
	* '''Scalar ~~multiplies~~''': If <math>f</math> is defined in an open interval containing <math>c</math> and is continuous at <math>c</math>, and <math>\lambda</math> is a real number, then <math>\lambda f</math> is continuous at <math>c</math>.		* '''Scalar multiples''': If <math>f</math> is defined in an open interval containing <math>c</math> and is continuous at <math>c</math>, and <math>\lambda</math> is a real number, then <math>\lambda f</math> is continuous at <math>c</math>.
	* '''Multiplicative identity''': The [[constant function]] that sends everything to <math>1</math> is continuous at <math>c</math>.		* '''Multiplicative identity''': The [[constant function]] that sends everything to <math>1</math> is continuous at <math>c</math>.
	* '''Multiplicative closure''': If <math>f,g</math> are functions defined in open intervals containing <math>c</math> and both of them are [[fact about::continuous function\|continuous]] at <math>c</math>, then the [[fact about::pointwise product of functions\|pointwise product]] <math>f \cdot g</math> is continuous at <math>c</math>.		* '''Multiplicative closure''': If <math>f,g</math> are functions defined in open intervals containing <math>c</math> and both of them are [[fact about::continuous function\|continuous]] at <math>c</math>, then the [[fact about::pointwise product of functions\|pointwise product]] <math>f \cdot g</math> is continuous at <math>c</math>.
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	* '''Additive closure''': If <math>f,g</math> are functions defined and continuous in open intervals containing <math>c</math>, then the [[fact about::pointwise sum of functions\|pointwise sum]] <math>f + g</math> is continuous at <math>c</math>.		* '''Additive closure''': If <math>f,g</math> are functions defined and continuous in open intervals containing <math>c</math>, then the [[fact about::pointwise sum of functions\|pointwise sum]] <math>f + g</math> is continuous at <math>c</math>.
	* '''Scalar ~~multiplies~~''': If <math>f</math> is defined and continuous in an open interval containing <math>c</math>>, and <math>\lambda</math> is a real number, then <math>\lambda f</math> is continuous at <math>c</math>.		* '''Scalar multiples''': If <math>f</math> is defined and continuous in an open interval containing <math>c</math>>, and <math>\lambda</math> is a real number, then <math>\lambda f</math> is continuous at <math>c</math>.
	* '''Multiplicative identity''': The [[constant function]] that sends everything to <math>1</math> is continuous in an open interval containing <math>c</math>.		* '''Multiplicative identity''': The [[constant function]] that sends everything to <math>1</math> is continuous in an open interval containing <math>c</math>.
	* '''Multiplicative closure''': If <math>f,g</math> are functions defined and continuous in open intervals containing <math>c</math>, then the [[fact about::pointwise product of functions\|pointwise product]] <math>f \cdot g</math> is continuous at <math>c</math>.		* '''Multiplicative closure''': If <math>f,g</math> are functions defined and continuous in open intervals containing <math>c</math>, then the [[fact about::pointwise product of functions\|pointwise product]] <math>f \cdot g</math> is continuous at <math>c</math>.

Vipul at 15:54, 16 October 2011

2011-10-16T15:54:23Z

@@ Line 5: / Line 5: @@
 Suppose <math>c \in \R</math>. Then, the following are true:
-* ''Additive closure'': If <math>f,g</math> are functions defined in open intervals containing <math>c</math> and both of them are [[fact about::continuous function|continuous]] at <math>c</math>, then the [[fact about::pointwise sum of functions|pointwise sum]] <math>f + g</math> is continuous at <math>c</math>.
+* '''Additive closure''': If <math>f,g</math> are functions defined in open intervals containing <math>c</math> and both of them are [[fact about::continuous function|continuous]] at <math>c</math>, then the [[fact about::pointwise sum of functions|pointwise sum]] <math>f + g</math> is continuous at <math>c</math>.
-* ''Scalar multiplies'': If <math>f</math> is defined in an open interval containing <math>c</math> and is continuous at <math>c</math>, and <math>\lambda</math> is a real number, then <math>\lambda f</math> is continuous at <math>c</math>.
+* '''Scalar multiplies''': If <math>f</math> is defined in an open interval containing <math>c</math> and is continuous at <math>c</math>, and <math>\lambda</math> is a real number, then <math>\lambda f</math> is continuous at <math>c</math>.
-* ''Multiplicative closure'': If <math>f,g</math> are functions defined in open intervals containing <math>c</math> and both of them are [[fact about::continuous function|continuous]] at <math>c</math>, then the [[fact about::pointwise product of functions|pointwise product]] <math>f \cdot g</math> is continuous at <math>c</math>.
+* '''Multiplicative identity''': The [[constant function]] that sends everything to <math>1</math> is continuous at <math>c</math>.

Vipul: Created page with "==Statement== ===Continuity at a point version=== Suppose $c \in \R$ . Then, the following are true: * ''Additive closure'': If $f,g$ are functions define..."

2011-10-16T15:44:38Z

Created page with "==Statement== ===Continuity at a point version=== Suppose <math>c \in \R</math>. Then, the following are true: * ''Additive closure'': If <math>f,g</math> are functions define..."

New page

==Statement==

===Continuity at a point version===

Suppose <math>c \in \R</math>. Then, the following are true:

* ''Additive closure'': If <math>f,g</math> are functions defined in open intervals containing <math>c</math> and both of them are [[fact about::continuous function|continuous]] at <math>c</math>, then the [[fact about::pointwise sum of functions|pointwise sum]] <math>f + g</math> is continuous at <math>c</math>.
* ''Scalar multiplies'': If <math>f</math> is defined in an open interval containing <math>c</math> and is continuous at <math>c</math>, and <math>\lambda</math> is a real number, then <math>\lambda f</math> is continuous at <math>c</math>.
* ''Multiplicative closure'': If <math>f,g</math> are functions defined in open intervals containing <math>c</math> and both of them are [[fact about::continuous function|continuous]] at <math>c</math>, then the [[fact about::pointwise product of functions|pointwise product]] <math>f \cdot g</math> is continuous at <math>c</math>.

Continuous functions form a unital algebra - Revision history

Vipul at 15:54, 16 October 2011

Vipul at 15:54, 16 October 2011

Vipul: Created page with "==Statement== ===Continuity at a point version=== Suppose c \in \R. Then, the following are true: * ''Additive closure'': If f,g are functions define..."

Vipul: Created page with "==Statement== ===Continuity at a point version=== Suppose $c \in \R$ . Then, the following are true: * ''Additive closure'': If $f,g$ are functions define..."