Difference between revisions of "Continuous functions form a vector space"

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==Statement==
 
==Statement==
  
Suppose <math>I</math> 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 <math>[a,b],[a,b),(a,b),(a,b],(-\infty,b),(-\infty,b],(a,\infty),[ainfty),(-\infty,\infty)</math>). A [[fact about::continuous function]] on <math>I</math> is a function on <math>I</math> that is continuous at all points on the ''interior'' of <math>I</math> and has the appropriate one-sided continuity at the boundary points (if they exist).
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Suppose <math>I</math> 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 <math>[a,b],[a,b),(a,b),(a,b],(-\infty,b),(-\infty,b],(a,\infty),[a,\infty),(-\infty,\infty)</math>). A [[fact about::continuous function]] on <math>I</math> is a function on <math>I</math> that is continuous at all points on the ''interior'' of <math>I</math> and has the appropriate one-sided continuity at the boundary points (if they exist).
  
 
The continuous functions on <math>I</math> form a [[real vector space]], in the sense that the following hold:
 
The continuous functions on <math>I</math> form a [[real vector space]], in the sense that the following hold:

Revision as of 15:33, 16 October 2011

Statement

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 f + g.
  • Scalar multiplies: If \lambda \in \R and f is a continuous function on I, then \lambda f is also a continuous function on I.