Darboux's theorem (analysis)

Darboux's theorem

Let I be an closed interval, f : I → R a real-valued differentiable function. Then f ′ has the intermediate value property: If a and b are points in I with a < b , then for every y between f ′ ( a ) and f ′ ( b ) , there exists an x in ( a , b ) such that f ′ ( x ) = y .

Proof

If y equals f ′ ( a ) or f ′ ( b ) , then setting x equal to a or b , respectively, works. Therefore, without loss of generality, we may assume that y is strictly between f ′ ( a ) and f ′ ( b ) , and in particular that f ′ ( a ) > y > f ′ ( b ) . Define a new function ϕ : I → R by ϕ ( t ) = f ( t ) − y t .

Since ϕ is continuous on the closed interval [ a , b ] , its maximum value on that interval is attained, according to the extreme value theorem, at a point x in that interval, i.e. at some x ∈ [ a , b ] . Because ϕ ′ ( a ) = f ′ ( a ) − y > y − y = 0 , we see ϕ cannot attain its maximum value on [ a , b ] at a . Because ϕ ′ ( b ) = f ′ ( b ) − y < y − y = 0 , we see ϕ cannot attain its maximum value on [ a , b ] at b . Therefore, x ∈ ( a , b ) . Hence, by Fermat's theorem, ϕ ′ ( x ) = 0 , i.e. f ′ ( x ) = y .

Another proof can be given by combining the mean value theorem and the intermediate value theorem.

In fact, let's take c = 1 2 ( a + b ) . For a ≤ t ≤ c , define α ( t ) = a and β ( t ) = 2 t − a . And for c ≤ t ≤ b , define α ( t ) = 2 t − b and β ( t ) = b .

Thus, for t ∈ ( a , b ) we have a ≤ α ( t ) < β ( t ) ≤ b . Now, define g ( t ) = ( f ∘ β ) ( t ) − ( f ∘ α ) ( t ) β ( t ) − α ( t ) with a < t < b . g is continuous in ( a , b ) .

Furthermore, g ( t ) → f ′ ( a ) when t → a and g ( t ) → f ′ ( b ) when t → b , and, therefore, from the Intermediate Value Theorem, if λ ∈ ( f ′ ( a ) , f ′ ( b ) ) then, there exists t 0 ∈ ( a , b ) such that g ( t 0 ) = λ . Let's fix t 0 .

From the Mean Value Theorem, there exists a point x ∈ ( α ( t 0 ) , β ( t 0 ) ) such that f ′ ( x ) = g ( t 0 ) . Hence, f ′ ( x ) = λ .

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 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 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 ) is a Darboux function that 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 f on the real line can be written as the sum of two Darboux functions. 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. Such functions exist and are Darboux but nowhere continuous.