Difference between revisions of "Intermediate value property"
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A [[real]] [[function]] is said to have the '''intermediate value property''' on an [[interval]] <math>[a, b]</math> if, for each value <math>v</math> between <math>\displaystyle f(a)</math> and <math>f(b)</math>, there is some <math>c \in (a, b)</math> such that <math>f(c) = v</math>. Thus, a function with the intermediate value property takes all intermediate values between any two points. | A [[real]] [[function]] is said to have the '''intermediate value property''' on an [[interval]] <math>[a, b]</math> if, for each value <math>v</math> between <math>\displaystyle f(a)</math> and <math>f(b)</math>, there is some <math>c \in (a, b)</math> such that <math>f(c) = v</math>. Thus, a function with the intermediate value property takes all intermediate values between any two points. | ||
| − | The simplest, and most important, examples of functions with this property are the [[continuous]] | + | The simplest, and most important, examples of functions with this property are the [[continuous function]]s. |
Revision as of 20:40, 8 November 2006
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A real function is said to have the intermediate value property on an interval ![$[a, b]$](http://latex.artofproblemsolving.com/d/a/2/da2e551d2ca2155b8d8f4935d2e9757722c9bab6.png)
if, for each value 
between 
and 
, there is some 
such that 
. Thus, a function with the intermediate value property takes all intermediate values between any two points.
The simplest, and most important, examples of functions with this property are the continuous functions.
