{{short description|All derivatives have the intermediate value property}} {{About|Darboux's theorem related to the intermediate value theorem|Darboux's theorem in differential geometry|Darboux's theorem}}
In real analysis, '''Darboux's theorem''' states that the derivative of any real-valued function of a real variable has the '''intermediate value property''', that is, that the image of an interval is also an interval.
When <math>f</math> is continuously differentiable, this is a consequence of the intermediate value theorem. But even when <math>f'</math> is ''not'' continuous, Darboux's theorem places a restriction on the behaviour of <math>f'</math> over any closed interval.
==Statement of the theorem== Let <math>I</math> be an open interval, and let <math>f\colon I\to \R</math> be 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<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="Apostol1974">Apostol, Tom M.: Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd edition, Addison-Wesley Longman, Inc. (1974), page 112.</ref><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><ref name="Rudin1976">Rudin, Walter: Principles of Mathematical Analysis, 3rd edition, MacGraw-Hill, Inc. (1976), page 108</ref>
The original proof by Jean Gaston Darboux has been published in 1875.<ref>{{citation |last = Darboux |first = Gaston |year = 1875 |title = Mémoire sur les fonctions discontinues | trans-title = Dissertation on discontinuous functions |lang = fr |journal = Annales Scientifiques de l'É.N.S., Serie 2 |volume = 4 |date = 1875 |publisher = École Normale Supérieure |publication-place = Paris |page = 109–110 |doi = 10.24033/asens.122}}</ref>
==Proofs== ===Proof from the extreme value theorem=== The first proof is based on the extreme value theorem.
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, gives the desired result. Now 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>. Let <math>\varphi\colon I\to \R</math> such that <math>\varphi(t)=f(t)-yt</math>. If it is the case that <math>f'(a)<y<f'(b)</math> we adjust our below proof, instead asserting that <math>\varphi</math> has its minimum on <math>[a,b]</math>.
Since <math>\varphi</math> is continuous on the closed interval <math>[a,b]</math>, the maximum value of <math>\varphi</math> on <math>[a,b]</math> is attained at some point in <math>[a,b]</math>, according to the extreme value theorem.
Because <math>\varphi'(a)=f'(a)-y> 0</math>, we know <math>\varphi</math> cannot attain its maximum value at <math>a</math>. (If it did, then <math> (\varphi(t)-\varphi(a))/(t-a) \leq 0 </math> for all <math> t \in (a,b] </math>, which implies <math> \varphi'(a) \leq 0 </math>.)
Likewise, because <math>\varphi'(b)=f'(b)-y<0</math>, we know <math>\varphi</math> cannot attain its maximum value at <math>b</math>.
Therefore, <math>\varphi</math> must attain its maximum value at some point <math>x\in(a,b)</math>. Hence, by Fermat's theorem, <math>\varphi'(x)=0</math>, i.e. <math>f'(x)=y</math>.
===Proof from the mean and intermediate value theorems=== The second proof is based on combining the mean value theorem and the intermediate value theorem.<ref name="Apostol1974"/><ref name="Olsen2004"/>
Define <math>c = \frac{1}{2} (a + b)</math>. For <math>a \leq t \leq c,</math> define <math>\alpha (t) = a</math> and <math>\beta (t) = 2t - a</math>. And for <math>c \leq t \leq b,</math> define <math>\alpha (t) = 2t - b</math> and <math>\beta(t) = b</math>.
Thus, for <math>t \in (a,b)</math> we have <math>a \leq \alpha (t) < \beta (t) \leq b</math>. Now, define <math>g(t) = \frac{(f \circ \beta)(t) - (f \circ \alpha)(t)}{\beta(t) - \alpha(t)}</math> with <math>a < t < b</math>. <math>\, g</math> is continuous in <math>(a, b)</math>.
Furthermore, <math>g(t) \rightarrow {f}' (a)</math> when <math>t \rightarrow a</math> and <math>g(t) \rightarrow {f}' (b)</math> when <math>t \rightarrow b</math>; therefore, from the Intermediate Value Theorem, if <math>y \in ({f}' (a), {f}' (b))</math> then, there exists <math>t_0 \in (a, b)</math> such that <math>g(t_0) = y</math>. Let's fix <math>t_0</math>.
From the Mean Value Theorem, there exists a point <math>x \in (\alpha (t_0), \beta (t_0))</math> such that <math>{f}'(x) = g(t_0)</math>. Hence, <math>{f}' (x) = y</math>.
==Darboux function== A '''Darboux function''' is a real-valued function <math>f</math> which has the "intermediate value property": for any two values <math>a</math> and <math>b</math> in the domain of <math>f</math>, and any <math>y</math> between <math>f(a)</math> and <math>f(b)</math>, there is some <math>c</math> between <math>a</math> and <math>b</math> with <math>y=f(c)</math>.<ref name=Cie>{{cite book | last=Ciesielski | first=Krzysztof | title=Set theory for the working mathematician | zbl=0938.03067 | series=London Mathematical Society Student Texts | volume=39 | location=Cambridge | publisher=Cambridge University Press | year=1997 | isbn=0-521-59441-3 | pages=106–111 }}</ref> By the intermediate value theorem, every continuous function on a real interval is a Darboux function. Darboux's contribution was to show that there are discontinuous Darboux functions.
Every discontinuity of a Darboux function is 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 topologist's sine curve function: :<math>x \mapsto \begin{cases}\sin(1/x) & \text{for } x\ne 0, \\ 0 &\text{for } x=0. \end{cases}</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 even though it is not continuous at one point.
An example of a Darboux function that is nowhere continuous is Conway's base 13 function. Another is Bergfeldt's function where a real number ''x'' is written in expanded in binary with digits <math>(x_i)_{i\in\mathbb Z_+}</math> each 0 or 1, and <math>f(x)=\sum\limits_{k=1}^\infty \frac{(-1)^{x_k}}{k}</math> if the series converges for that ''x'' and 0 if it does not.<ref>{{cite web | url = https://math.stackexchange.com/questions/75589/open-maps-which-are-not-continuous/2933144#2933144 | title = Open maps which are not continuous | website = Stack Exchange Mathematics | date = 2018-09-27 | access-date = 2023-07-10 | at = In an answer to the question | author = Bergfeldt, Aksel}}</ref>
Darboux functions are a quite general class of functions. It turns out that any real-valued function ''ƒ'' 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.
A '''strongly Darboux function''' is one for which the image of every (non-empty) open interval is the whole real line.<ref name=Cie/>
== Further restrictions on derivatives ==
Darboux's theorem gives a necessary condition for a function to be a derivative, but it is not sufficient. Every derivative of a real function is also of Baire class one, and the set of points at which a derivative is discontinuous is a meagre <math>F_\sigma</math> set. Conversely, every meagre <math>F_\sigma</math> subset of the real line can occur as the discontinuity set of a derivative.<ref>{{cite journal |last1=Bruckner |first1=Andrew M. |last2=Leonard |first2=J. L. |title=Derivatives |journal=American Mathematical Monthly |volume=73 |issue=4, Part II |year=1966 |pages=24–56}}</ref>
A finer restriction is on the sublevel sets of a derivative. For a real function <math>f</math>, its associated superlevel and sublevel sets are <math>\{x:f(x)>a\}</math> and <math>\{x:f(x)<a\}</math>, where <math>a</math> is real. Zahorski introduced classes <math>M_0,\ldots,M_5</math> of sets describing how large such associated sets must be near their own points. In this terminology, one has the following theorems: * Every finite derivative has associated sets in <math>M_3</math>. * Every bounded derivative has associated sets in <math>M_4</math>. Moreover, a set is an associated set of some bounded derivative if and only if it belongs to <math>M_4</math>.<ref name="Bruckner1994">{{cite book |last=Bruckner |first=Andrew M. |title=Differentiation of Real Functions |edition=2nd |series=CRM Monograph Series |volume=5 |publisher=American Mathematical Society |year=1994 |isbn=0-8218-6990-6 |pages=61–67}}</ref>
Intuitively, if <math>f=F'</math> and <math>f(x_0)>a</math>, then the set on which <math>f>a</math> cannot be arbitrarily sparse near <math>x_0</math>. If <math>f</math> is continuous at <math>x_0</math>, this is trivial: <math>f>a</math> throughout some neighbourhood of <math>x_0</math>, so the local density is <math>1</math>. The Zahorski conditions express weaker density requirements that remain valid even when the derivative is discontinuous.
More explicitly, a non-empty <math>F_\sigma</math> set <math>E</math> belongs to <math>M_3</math> if, for every <math>x\in E</math>, any sequence of closed intervals <math>I_n</math> not containing <math>x</math>, with <math>\operatorname{dist}(x,I_n)\to 0</math> and <math>\lambda(I_n\cap E)=0</math>, satisfies :<math>\frac{\lambda(I_n)}{\operatorname{dist}(x,I_n)}\to 0,</math> where <math>\lambda</math> denotes Lebesgue measure. Thus, near a point of <math>E</math>, gaps in <math>E</math> cannot have length comparable to their distance from the point. The class <math>M_4</math> is stronger: <math>E</math> belongs to <math>M_4</math> if it can be written as a countable union of closed sets <math>E=\bigcup K_n</math> such that, on each <math>K_n</math>, the set <math>E</math> occupies a uniformly positive proportion of every sufficiently small one-sided interval whose length is comparable with its distance from the point. In this sense, <math>M_3</math> rules out large nearby holes, while <math>M_4</math> imposes a uniform positive lower-density condition.
==Notes== {{Reflist}}
==External links== * {{PlanetMath attribution|id=3055|title=Darboux's theorem}} * {{SpringerEOM|title=Darboux theorem|id=p/d030190}}
Category:Theorems in calculus Category:Theory of continuous functions Category:Theorems in real analysis Category:Articles containing proofs