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 "C"1(["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.
Darboux's theorem.
Let formula_1 be a closed interval, formula_2 be a real-valued differentiable function. Then formula_3 has the intermediate value property: If formula_4 and formula_5 are points in formula_1 with formula_7 between formula_8 and formula_9, there exists an formula_10 in formula_11 such that formula_12.
Proofs.
Proof 1. The first proof is based on the extreme value theorem.
If formula_13 equals formula_8 or formula_9, then setting formula_10 equal to formula_4 or formula_5, respectively, gives the desired result. Now assume that formula_13 is strictly between formula_8 and formula_9, and in particular that formula_22. Let formula_23 such that formula_24. If it is the case that formula_25 we adjust our below proof, instead asserting that formula_26 has its minimum on formula_11.
Since formula_26 is continuous on the closed interval formula_11, the maximum value of formula_26 on formula_11 is attained at some point in formula_11, according to the extreme value theorem.
Because formula_33, we know formula_26 cannot attain its maximum value at formula_4. (If it did, then formula_36 for all formula_37, which implies formula_38.)
Likewise, because formula_39, we know formula_26 cannot attain its maximum value at formula_5.
Therefore, formula_26 must attain its maximum value at some point formula_43. Hence, by Fermat's theorem, formula_44, i.e. formula_12.
Proof 2. The second proof is based on combining the mean value theorem and the intermediate value theorem.
Define formula_46.
For formula_47 define formula_48 and formula_49.
And for formula_50 define formula_51 and formula_52.
Thus, for formula_53 we have formula_54.
Now, define formula_55 with formula_56.
formula_57 is continuous in formula_58.
Furthermore, formula_59 when formula_60 and formula_61 when formula_62; therefore, from the Intermediate Value Theorem, if formula_63 then, there exists formula_64 such that formula_65.
Let's fix formula_66.
From the Mean Value Theorem, there exists a point formula_67 such that formula_68.
Hence, formula_69.
Darboux function.
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". 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:
formula_70
By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function formula_71 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. 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. The Conway base 13 function is again an example.
Notes.
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