Continuous functions form a unital algebra: Difference between revisions

Latest revision as of 15:54, 16 October 2011

Statement

Continuity at a point version

Suppose $c\in \mathbb {R}$ . Then, the following are true:

Additive closure: If $f,g$ are functions defined in open intervals containing $c$ and both of them are continuous at $c$ , then the pointwise sum $f+g$ is continuous at $c$ .
Scalar multiples: If $f$ is defined in an open interval containing $c$ and is continuous at $c$ , and $\lambda$ is a real number, then $\lambda f$ is continuous at $c$ .
Multiplicative identity: The constant function that sends everything to $1$ is continuous at $c$ .
Multiplicative closure: If $f,g$ are functions defined in open intervals containing $c$ and both of them are continuous at $c$ , then the pointwise product $f\cdot g$ is continuous at $c$ .

There is a technical way of forming a unital algebra (a vector space endowed with an identity and a compatible multiplication) from the set of continuous functions at the point $c$ , once we identify (as being the same) any two functoins that agree on an open interval containing $c$ . This approach is called taking the germ of the function.

Continuity around a point version

Suppose $c\in \mathbb {R}$ . Then, the following are true:

Additive closure: If $f,g$ are functions defined and continuous in open intervals containing $c$ , then the pointwise sum $f+g$ is continuous at $c$ .
Scalar multiples: If $f$ is defined and continuous in an open interval containing $c$ >, and $\lambda$ is a real number, then $\lambda f$ is continuous at $c$ .
Multiplicative identity: The constant function that sends everything to $1$ is continuous in an open interval containing $c$ .
Multiplicative closure: If $f,g$ are functions defined and continuous in open intervals containing $c$ , then the pointwise product $f\cdot g$ is continuous at $c$ .

There is a technical way of forming a unital algebra (a vector space endowed with an identity and a compatible multiplication) from the set of continuous functions at the point $c$ , once we identify (as being the same) any two functoins that agree on an open interval containing $c$ . This approach is called taking the germ of the function.

Continuity on an interval version

Suppose $I$ is an interval (possibly open, closed, or infinite from the left side and possibly open, closed, or infinite from the right side -- so it could be of the form $[a,b],[a,b),(a,b),(a,b],(-\infty ,b),(-\infty ,b],(a,\infty ),[a,\infty ),(-\infty ,\infty )$ ). A continuous function on $I$ is a function on $I$ that is continuous at all points on the interior of $I$ and has the appropriate one-sided continuity at the boundary points (if they exist).

The continuous functions on $I$ form a real vector space, in the sense that the following hold:

Additive: A sum of continuous functions is continuous: If $f,g$ are both continuous functions on $I$ , so is the pointwise sum of functions $f+g$ .
Scalar multiples: If $\lambda \in \mathbb {R}$ and $f$ is a continuous function on $I$ , then $\lambda f$ is also a continuous function on $I$ .
Multiplicative identity': The constant function sending everything to 1 is a continuous function on $I$ .
Multiplicative closure: A product of continuous functions is continuous: If $f,g$ are both continuous functions on $I$ , so is the pointwise product of functions $f\cdot g$ .

Facts used

Continuous functions form a vector space: This takes care of the addition and scalar multiples aspects
Limit is multiplicative: This takes care of multiplicative closure

@@ Line 6: / Line 6: @@
 * '''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 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>.
@@ Line 17: / Line 17: @@
 * '''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 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>.