Intermediate value theorem

In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, the intermediate value theorem states that if 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a <a href="/facts/Continuous_function/sKbl02pB">continuous</a> <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> whose <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> contains the <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> [a, b], then it takes on any given value between 
 
 
 
 f
 (
 a
 )
 
 
 {\displaystyle f(a)}
 
 and 
 
 
 
 f
 (
 b
 )
 
 
 {\displaystyle f(b)}
 
 at some point within the interval.
This has two important <a href="/facts/Corollary/ntQ8zR9N">corollaries</a>:

<ol><li>If a continuous function has values of opposite sign inside an interval, then it has a <a href="/facts/Zero_of_a_function/mlK3dhoT">root</a> in that interval (Bolzano's theorem).</li>
<li>The <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of a continuous function over an interval is itself an interval.</li></ol>

Intermediate value theorem open-in-new

Intermediate value theorem