| Darboux's Theorem (analysis) |
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Darboux's theorem is a of an Interval is also an interval. Note that when ''f'' is Continuously differentiable (''f'' in ''C''1( {Link without Title} )), this is trivially true by the Intermediate Value Theorem . But even when is ''not'' continuous, Darboux's theorem places a severe restriction on what it can be. DARBOUX'S THEOREM Let ''f'' : → R be a real-valued Continuous Function on [''a'',''b'' , which is Differentiable on (''a'',''b''), differentiable from the right at ''a'', and differentiable from the left at ''b''. Then satisfies the intermediate value property: for every ''t'' between and , there is some ''x'' in [''a'',''b''] such that . PROOF Without loss of generality we migth and shall assume . Let ''g''(''x'') := ''f''(''x'') - ''tx''. Then , , and we wish to find a zero of . Since ''g'' is a continuous function on by the Extreme Value Theorem it attains a Maximum on [''a'',''b'' . This maximum cannot be at ''a'', since so ''g'' is locally increasing at ''a''. Similarly, , so ''g'' is locally decreasing at ''b'' and cannot have a maximum at ''b''. So the maximum is attained at some ''c'' in (''a'',''b''). But then by Fermat's Theorem (stationary Points) . SEE ALSO EXTERNAL LINKS |
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