Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Darboux's theorem (analysis)
open-in-new
<p>In mathematics, Darboux's theorem is a <a href="/facts/Theorem/f2Pe1FMD">theorem</a> in <a href="/facts/Real_analysis/12WjlXCW">real analysis</a>, named after <a href="/facts/Jean_Gaston_Darboux/evh2nMHN">Jean Gaston Darboux</a>. It states that every function that results from the <a href="/facts/Derivative/7Ito70Dh">differentiation</a> of another function has the intermediate value property: the <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of an <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> is also an interval. </p><p>When <i>ƒ</i> is <a href="/facts/Continuously_differentiable/jQqTgLk1">continuously differentiable</a> (<i>ƒ</i> in <i>C</i>1([<i>a</i>,<i>b</i>])), this is a consequence of the <a href="/facts/Intermediate_value_theorem/SaFdYE7W">intermediate value theorem</a>. But even when <i>ƒ′</i> is <i>not</i> continuous, Darboux's theorem places a severe restriction on what it can be. </p>