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'Darboux's theorem' is a theorem in real analysis, named after Jean Gaston Darboux. It states that all functions which result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval.
Note that when ''f'' is continuously differentiable (''f'' in ''C''¹([''a'',''b''])), this is trivially true by the intermediate value theorem. But even when $f\text{'}$ is ''not'' continuous, Darboux's theorem places a severe restriction on what it can be.

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Darboux's theorem

Proof

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Darboux's theorem

Let ''f'' : [''a'',''b''] → '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 $f\text{'}$ satisfies the intermediate value property: for every ''t'' between $f\text{'}\_\{+\}(a)$ and $f\text{'}\_\{-\}(b)$, there is some ''x'' in [''a'',''b''] such that $f\text{'}(x)\; =\; t$.

Proof

Without loss of generality we might and shall assume $f\text{'}\_\{+\}(a)\; >\; t\; >\; f\text{'}\_\{-\}(b)$. Let ''g''(''x'') := ''f''(''x'') - ''tx''. Then $g\text{'}(x)\; =\; f\text{'}(x)\; -\; t$, $g\text{'}\_\{+\}(a)\; >\; 0\; >\; g\text{'}\_\{-\}(b)$, and we wish to find a zero of $g\text{'}$.
Since ''g'' is a continuous function on [''a'',''b''], by the extreme value theorem it attains a maximum on [''a'',''b'']. This maximum cannot be at ''a'', since $g\text{'}\_\{+\}(a)\; >\; 0$ 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 $g\text{'}(c)\; =\; 0$ by Fermat's theorem (stationary points).

See also

★ Jean Gaston Darboux

★ Intermediate value theorem

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