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| '''Darboux's theorem''' (also known as the '''Intermediate Value Theorem'''<ref>{{cite book|last=Larson|first=Ron|title=Calculus of a single variable|year=2006|publisher=Houghton Mifflin Co.|location=Boston, Mass.|isbn=0618503048|pages=77|edition=8th ed.|coauthors=Robert P. Hostetler, Bruce H.|chapter=Continuity and One-Sided Limits}}</ref>) is a [[theorem]] in [[real analysis]], named after [[Jean Gaston Darboux]]. It states that all functions that result from the [[derivative|differentiation]] of other functions have the '''intermediate value property''': the [[image (mathematics)|image]] of an [[interval (mathematics)|interval]] is also an interval.
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| When ''f'' is [[continuous function|continuously]] differentiable (''f'' in ''C''<sup>1</sup>([''a'',''b''])), this is a consequence of the [[intermediate value theorem]]. But even when ''f′'' is ''not'' continuous, Darboux's theorem places a severe restriction on what it can be.
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| ==Darboux's theorem==
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| Let <math>I</math> be an [[open interval]], <math>f\colon I\to \R</math> a real-valued differentiable function. Then <math>f'</math> has the '''intermediate value property''': If <math>a</math> and <math>b</math> are points in <math>I</math> with <math>a\leq b</math>, then for every <math>y</math> between <math>f'(a)</math> and <math>f'(b)</math>, there exists an <math>x</math> in <math>[a,b]</math> such that <math>f'(x)=y</math>.<ref name="Olsen2004"/>
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| ==Proof==
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| If <math>y</math> equals <math>f'(a)</math> or <math>f'(b)</math>, then setting <math>x</math> equal to <math>a</math> or <math>b</math>, respectively, works. Therefore, without loss of generality, we may assume that <math>y</math> is strictly between <math>f'(a)</math> and <math>f'(b)</math>, and in particular that <math>f'(a)>y>f'(b)</math>. Define a new function <math>\phi\colon I\to \R</math> by
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| :<math>\phi(t)=f(t)-yt.</math>
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| Since <math>\phi</math> is continuous on the closed interval <math>[a,b]</math>, its maximum value on that interval is attained, according to the [[extreme value theorem]], at a point <math>x</math> in that interval, i.e. at some <math>x\in[a,b]</math>. Because <math>\phi'(a)=f'(a)-y>y-y=0</math> and <math>\phi'(b)=f'(b)-y<y-y=0</math>, [[Fermat's theorem (stationary points)|Fermat's theorem]] implies that neither <math>a</math> nor <math>b</math> can be a point, such as <math>x</math>, at which <math>\phi</math> attains a local maximum. Therefore, <math>x\in(a,b)</math>. Hence, again by Fermat's theorem, <math>\phi'(x)=0</math>, i.e. <math>f'(x)=y</math>.<ref name="Olsen2004">Olsen, Lars: ''A New Proof of Darboux's Theorem'', Vol. 111, No. 8 (Oct., 2004) (pp. 713-715), The American Mathematical Monthly</ref>
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| Another proof based solely on the [[mean value theorem]] and the [[intermediate value theorem]] is due to Lars Olsen.<ref name="Olsen2004"/>
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| ==Darboux function==
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| A '''Darboux function''' is a [[real-valued function]] ''f'' which has the "intermediate value property": for any two values ''a'' and ''b'' in the domain of ''f'', and any ''y'' between ''f''(''a'') and ''f''(''b''), there is some ''c'' between ''a'' and ''b'' with ''f''(''c'') = ''y''. By the [[intermediate value theorem]], every [[continuous function]] is a Darboux function. Darboux's contribution was to show that there are discontinuous Darboux functions.
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| Every [[discontinuity (mathematics)|discontinuity]] of a Darboux function is [[essential discontinuity|essential]], that is, at any point of discontinuity, at least one of the left hand and right hand limits does not exist.
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| An example of a Darboux function that is discontinuous at one point, is the function <math>x \mapsto \sin(1/x)</math>.
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| By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function <math>x \mapsto x^2\sin(1/x)</math> is a Darboux function that is not continuous.
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| An example of a Darboux function that is [[Nowhere continuous function|nowhere continuous]] is the [[Conway base 13 function]].
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| Darboux functions are a quite general class of functions. It turns out that any real-valued function ''f'' on the real line can be written as the sum of two Darboux functions.<ref>Bruckner, Andrew M: ''Differentiation of real functions'', 2 ed, page 6, American Mathematical Society, 1994</ref> This implies in particular that the class of Darboux functions is not closed under addition.
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| ==Notes==
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| <references/>
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| ==External links==
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| * {{PlanetMath attribution|id=3055|title=Darboux's theorem}}
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| * {{springer|title=Darboux theorem|id=p/d030190}}
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| [[Category:Theorems in calculus]]
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| [[Category:Continuous mappings]]
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| [[Category:Theorems in real analysis]]
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| [[Category:Articles containing proofs]]
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