|
|
| Line 1: |
Line 1: |
| '''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.
| | The name of the author is Jayson. Distributing manufacturing is how he makes a residing. Alaska is where I've always been residing. The favorite hobby for him and his children is to play lacross and he would by no means give it up.<br><br>Here is my site ... telephone psychic ([http://www.010-5260-5333.com/index.php?document_srl=1880&mid=board_ALMP66 click the next internet site]) |
| | |
| 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.
| |
| | |
| ==Darboux's theorem==
| |
| 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"/>
| |
| | |
| ==Proof==
| |
| 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
| |
| :<math>\phi(t)=f(t)-yt.</math>
| |
| 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>
| |
| | |
| Another proof based solely on the [[mean value theorem]] and the [[intermediate value theorem]] is due to Lars Olsen.<ref name="Olsen2004"/>
| |
| | |
| ==Darboux function==
| |
| 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.
| |
| | |
| 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.
| |
| | |
| An example of a Darboux function that is discontinuous at one point, is the function <math>x \mapsto \sin(1/x)</math>.
| |
| | |
| 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.
| |
| | |
| An example of a Darboux function that is [[Nowhere continuous function|nowhere continuous]] is the [[Conway base 13 function]].
| |
| | |
| 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.
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| ==External links==
| |
| * {{PlanetMath attribution|id=3055|title=Darboux's theorem}}
| |
| * {{springer|title=Darboux theorem|id=p/d030190}}
| |
| | |
| [[Category:Theorems in calculus]]
| |
| [[Category:Continuous mappings]]
| |
| [[Category:Theorems in real analysis]]
| |
| [[Category:Articles containing proofs]]
| |
The name of the author is Jayson. Distributing manufacturing is how he makes a residing. Alaska is where I've always been residing. The favorite hobby for him and his children is to play lacross and he would by no means give it up.
Here is my site ... telephone psychic (click the next internet site)