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All derivatives have the intermediate value property
In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval.
When ƒ is continuously differentiable (ƒ in C1([a,b])), this is a consequence of the intermediate value theorem. But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be.
Let I {\displaystyle I} be a closed interval, f : I → R {\displaystyle f\colon I\to \mathbb {R} } be a real-valued differentiable function. Then f ′ {\displaystyle f'} has the intermediate value property: If a {\displaystyle a} and b {\displaystyle b} are points in I {\displaystyle I} with a < b {\displaystyle a<b} , then for every y {\displaystyle y} between f ′ ( a ) {\displaystyle f'(a)} and f ′ ( b ) {\displaystyle f'(b)} , there exists an x {\displaystyle x} in [ a , b ] {\displaystyle [a,b]} such that f ′ ( x ) = y {\displaystyle f'(x)=y} .[1][2][3]
Proof 1. The first proof is based on the extreme value theorem.
If y {\displaystyle y} equals f ′ ( a ) {\displaystyle f'(a)} or f ′ ( b ) {\displaystyle f'(b)} , then setting x {\displaystyle x} equal to a {\displaystyle a} or b {\displaystyle b} , respectively, gives the desired result. Now assume that y {\displaystyle y} is strictly between f ′ ( a ) {\displaystyle f'(a)} and f ′ ( b ) {\displaystyle f'(b)} , and in particular that f ′ ( a ) > y > f ′ ( b ) {\displaystyle f'(a)>y>f'(b)} . Let φ : I → R {\displaystyle \varphi \colon I\to \mathbb {R} } such that φ ( t ) = f ( t ) − y t {\displaystyle \varphi (t)=f(t)-yt} . If it is the case that f ′ ( a ) < y < f ′ ( b ) {\displaystyle f'(a)<y<f'(b)} we adjust our below proof, instead asserting that φ {\displaystyle \varphi } has its minimum on [ a , b ] {\displaystyle [a,b]} .
Since φ {\displaystyle \varphi } is continuous on the closed interval [ a , b ] {\displaystyle [a,b]} , the maximum value of φ {\displaystyle \varphi } on [ a , b ] {\displaystyle [a,b]} is attained at some point in [ a , b ] {\displaystyle [a,b]} , according to the extreme value theorem.
Because φ ′ ( a ) = f ′ ( a ) − y > 0 {\displaystyle \varphi '(a)=f'(a)-y>0} , we know φ {\displaystyle \varphi } cannot attain its maximum value at a {\displaystyle a} . (If it did, then ( φ ( t ) − φ ( a ) ) / ( t − a ) ≤ 0 {\displaystyle (\varphi (t)-\varphi (a))/(t-a)\leq 0} for all t ∈ ( a , b ] {\displaystyle t\in (a,b]} , which implies φ ′ ( a ) ≤ 0 {\displaystyle \varphi '(a)\leq 0} .)
Likewise, because φ ′ ( b ) = f ′ ( b ) − y < 0 {\displaystyle \varphi '(b)=f'(b)-y<0} , we know φ {\displaystyle \varphi } cannot attain its maximum value at b {\displaystyle b} .
Therefore, φ {\displaystyle \varphi } must attain its maximum value at some point x ∈ ( a , b ) {\displaystyle x\in (a,b)} . Hence, by Fermat's theorem, φ ′ ( x ) = 0 {\displaystyle \varphi '(x)=0} , i.e. f ′ ( x ) = y {\displaystyle f'(x)=y} .
Proof 2. The second proof is based on combining the mean value theorem and the intermediate value theorem.[1][2]
Define c = 1 2 ( a + b ) {\displaystyle c={\frac {1}{2}}(a+b)} . For a ≤ t ≤ c , {\displaystyle a\leq t\leq c,} define α ( t ) = a {\displaystyle \alpha (t)=a} and β ( t ) = 2 t − a {\displaystyle \beta (t)=2t-a} . And for c ≤ t ≤ b , {\displaystyle c\leq t\leq b,} define α ( t ) = 2 t − b {\displaystyle \alpha (t)=2t-b} and β ( t ) = b {\displaystyle \beta (t)=b} .
Thus, for t ∈ ( a , b ) {\displaystyle t\in (a,b)} we have a ≤ α ( t ) < β ( t ) ≤ b {\displaystyle a\leq \alpha (t)<\beta (t)\leq b} . Now, define g ( t ) = ( f ∘ β ) ( t ) − ( f ∘ α ) ( t ) β ( t ) − α ( t ) {\displaystyle g(t)={\frac {(f\circ \beta )(t)-(f\circ \alpha )(t)}{\beta (t)-\alpha (t)}}} with a < t < b {\displaystyle a<t<b} . g {\displaystyle \,g} is continuous in ( a , b ) {\displaystyle (a,b)} .
Furthermore, g ( t ) → f ′ ( a ) {\displaystyle g(t)\rightarrow {f}'(a)} when t → a {\displaystyle t\rightarrow a} and g ( t ) → f ′ ( b ) {\displaystyle g(t)\rightarrow {f}'(b)} when t → b {\displaystyle t\rightarrow b} ; therefore, from the Intermediate Value Theorem, if y ∈ ( f ′ ( a ) , f ′ ( b ) ) {\displaystyle y\in ({f}'(a),{f}'(b))} then, there exists t 0 ∈ ( a , b ) {\displaystyle t_{0}\in (a,b)} such that g ( t 0 ) = y {\displaystyle g(t_{0})=y} . Let's fix t 0 {\displaystyle t_{0}} .
From the Mean Value Theorem, there exists a point x ∈ ( α ( t 0 ) , β ( t 0 ) ) {\displaystyle x\in (\alpha (t_{0}),\beta (t_{0}))} such that f ′ ( x ) = g ( t 0 ) {\displaystyle {f}'(x)=g(t_{0})} . Hence, f ′ ( x ) = y {\displaystyle {f}'(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.[4] 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 ↦ x 2 sin ( 1 / x ) {\displaystyle x\mapsto x^{2}\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 Conway's base 13 function. Another is Bergfeldt's function where a real number x is written in expanded in binary with digits ( x i ) i ∈ Z + {\displaystyle (x_{i})_{i\in \mathbb {Z} _{+}}} each 0 or 1, and f ( x ) = ∑ i = 1 ∞ ( − 1 ) x k k {\displaystyle f(x)=\sum \limits _{i=1}^{\infty }{\frac {(-1)^{x_{k}}}{k}}} if the series converges for that x and 0 if it does not.[5]
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.[6] 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. [4]
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