Darboux's theorem (analysis)#Darboux function

Let $I$ be a closed interval, $f\backslash colon\; I\backslash to\; \backslash R$ be a real-valued differentiable function. Then $f\text{'}$ has the intermediate value property: If $a$ and $b$ are points in $I$ with A New Proof of Darboux's Theorem, Vol. 111, No. 8 (Oct., 2004) (pp. 713–715), The American Mathematical MonthlyRudin, Walter: Principles of Mathematical Analysis, 3rd edition, MacGraw-Hill, Inc. (1976), page 108

Proof 1. The first proof is based on the extreme value theorem.

If $y$ equals $f\text{'}(a)$ or $f\text{'}(b)$, then setting $x$ equal to $a$ or $b$, respectively, gives the desired result. Now assume that $y$ is strictly between $f\text{'}(a)$ and $f\text{'}(b)$, and in particular that $f\text{'}(a)>y>f\text{'}(b)$. Let $\backslash varphi\backslash colon\; I\backslash to\; \backslash R$ such that $\backslash varphi(t)=f(t)-yt$. If it is the case that

Since $\backslash varphi$ is continuous on the closed interval $[a,b]$, the maximum value of $\backslash varphi$ on $[a,b]$ is attained at some point in $[a,b]$, according to the extreme value theorem.

Because $\backslash varphi\text{'}(a)=f\text{'}(a)-y>\; 0$, we know $\backslash varphi$ cannot attain its maximum value at $a$. (If it did, then $(\backslash varphi(t)-\backslash varphi(a))/(t-a)\; \backslash leq\; 0$ for all $t\; \backslash in\; (a,b]$, which implies $\backslash varphi\text{'}(a)\; \backslash leq\; 0$.)

Likewise, because , we know $\backslash varphi$ cannot attain its maximum value at $b$.

Therefore, $\backslash varphi$ must attain its maximum value at some point $x\backslash in(a,b)$. Hence, by Fermat's theorem, $\backslash varphi\text{'}(x)=0$, i.e. $f\text{'}(x)=y$.

Proof 2. The second proof is based on combining the mean value theorem and the intermediate value theorem.

Define $c\; =\; \backslash frac\{1\}\{2\}\; (a\; +\; b)$.

For $a\; \backslash leq\; t\; \backslash leq\; c,$ define $\backslash alpha\; (t)\; =\; a$ and $\backslash beta\; (t)\; =\; 2t\; -\; a$.

And for $c\; \backslash leq\; t\; \backslash leq\; b,$ define $\backslash alpha\; (t)\; =\; 2t\; -\; b$ and $\backslash beta(t)\; =\; b$.

Thus, for $t\; \backslash in\; (a,b)$ we have .

Now, define $g(t)\; =\; \backslash frac\{(f\; \backslash circ\; \backslash beta)(t)\; -\; (f\; \backslash circ\; \backslash alpha)(t)\}\{\backslash beta(t)\; -\; \backslash alpha(t)\}$ with .

$\backslash ,\; g$ is continuous in $(a,\; b)$.

Furthermore, $g(t)\; \backslash rightarrow\; \{f\}\text{'}\; (a)$ when $t\; \backslash rightarrow\; a$ and $g(t)\; \backslash rightarrow\; \{f\}\text{'}\; (b)$ when $t\; \backslash rightarrow\; b$; therefore, from the Intermediate Value Theorem, if $y\; \backslash in\; (\{f\}\text{'}\; (a),\; \{f\}\text{'}\; (b))$ then, there exists $t\_0\; \backslash in\; (a,\; b)$ such that $g(t\_0)\; =\; y$.

Let's fix $t\_0$.

From the Mean Value Theorem, there exists a point $x\; \backslash in\; (\backslash alpha\; (t\_0),\; \backslash beta\; (t\_0))$ such that $\{f\}\text{'}(x)\; =\; g(t\_0)$.

Hence, $\{f\}\text{'}\; (x)\; =\; y$.

A Darboux function is a real-valued function ƒ which has the "intermediate value property": for any two values a and b in the domain of ƒ, and any y between ƒ(a) and ƒ(b), there is some c between a and b with ƒ(c) = y.{{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 }} 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:

:

By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function $x\; \backslash mapsto\; x^2\backslash sin(1/x)$ is a Darboux function even though it is not continuous at one point.

An example of a Darboux function that is 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 ƒ on the real line can be written as the sum of two Darboux functions.Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994 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.

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• {{PlanetMath attribution|id=3055|title=Darboux's theorem}}
• {{SpringerEOM|title=Darboux theorem|id=p/d030190}}

Category:Theorems in calculus

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