Darboux’s Theorem (also called Darboux Continuity or Darboux Property) tells us that any derivative has the Intermediate Value Theorem property, even discontinuous ones. The IVT says that if a continuous function takes on a negative value and then switches to a positive value, it must take on a value of “0” somewhere in between.
Up until French mathematician Jean Gaston Darboux developed the proof of his theorem in 1975, it was widely believed that the IVT implied continuity. Darboux showed this wasn’t the case, demonstrating several differentiable functions with discontinuous derivatives [1].
Darboux made several contributions to the theory of singularities in differential equations. The similarly-named “Darboux theorem” concerns neighborhoods of points in manifolds [2].
Formal Definition of Darboux’s Theorem
Darboux’s Theorem is formally stated as [3]:
Suppose a function is differentiable on a closed interval [a, b] so that:
-
- f′(a) = y1
and
- f′(b) = y2
If d lies between y1 and y2, then there is a point c in the interval [a, b] with f′(c) = d.
Darboux’s Theorem Example
The above graph shows the function f(x) = x2 sin(1/x) and its discontinuous (at zero) derivative f′(x) = 2x sin(1/x) – cos(1/x). The problem with the derivative is that the second part, cos(1/x), isn’t defined at zero. But we can assign a value by tweaking the function’s definition:
This perfectly valid definition means that the derivative now has the IVT property for certain intervals [4]. For example [0.5, 2]. But it isn’t continuous, and behaves pathologically as it nears zero.
Many different proofs can be found in the literature. Several proofs of the theorem can be found in Dr. Mukta Bhandar’s paper Another Proof of Darboux’s Theorem[5] available here.
Darboux Property
The Darboux property is another name for Darboux’s theorem. However, some authors may use a slightly different definitions, which means that the two terms may in some cases also differ. For example, Beni Bogoşel states [6] that a function has the Darboux property if, for any interval I ⊂ ℝ, f(I) is also an interval; Bogoşel clarified in a comment that he was using a seldom-used form of the intermediate value property where the entire image is an interval. Continuity implies the Darboux property and the Darboux property implies the intermediate value property.
References
[1] Olsen, L. A New Proof of Darboux’s Theorem. The American Mathematical Monthly Vol. 111, No. 8 (Oct., 2004), pp. 713-715. Retrieved April 14, 2021 from: https://www.jstor.org/stable/4145046?seq=1
[2] Lesfari, A. Moser’s lemma and the Darboux Theorem. Universal Journal of Applied Mathematics 2(1): 36-39, 2014. Retrieved April 14, 2021 from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1048.8405&rep=rep1&type=pdf
[3] Larson, R. & Edwards, B. (2016). Calculus, 10th Edition. Cengage Learning.
[4] Math 10850, Honors Calculus 1. (2018). Retrieved April 14, 2021 from: https://www3.nd.edu/~dgalvin1/10850/10850_F18/10850-tutorial_12.pdf
[5] Bhandar, M. Another Proof of Darboux’s Theorem. Retrieved April 14, 2021 from: https://arxiv.org/pdf/1601.02719.pdf
[6] Bogoşel, B. Problem list. Retrieved March 3, 2023 from: https://mathproblems123.wordpress.com/more-than-problems/problems/
