A function $f\colon\mathbb R\to\mathbb R$ is called Darboux function, function with Darboux property or function with intermediate value property for for any $a<b$ and any $z$ between $f(a)$ and $f(b)$ there is $c$ between $a$ and $b$ such that $f(c)=z$. I.e. for any interval $I=(a,b)$ we have that $f[I]$ contains all points between $f(a)$ and $f(b)$.

A natural strengthening of this notion is the require that $f[I]=\mathbb R$ for every non-trivial interval $I$. In the book Ciesielski: Set Theory for the Working Mathematician p.106 such functions are called strongly Darboux.

However, I did not find many hits on strongly Darboux functions in Google Books or Google Scholar. One possible reason might be that such functions have been studied under a different name, too.

Are different names used for functions with the property that $f[I]=\mathbb R$ for every non-trivial interval $I$?


1 Answer 1


I suspect Ciesielski's term is recent (last 20 years or so) and of limited use among his students and co-workers, and it probably isn't generally known under another term, but I'll look through some of my things in the next couple of days and see what I can uncover about possible different terms.

For now, the following 8 February 2009 sci.math post of mine might be of interest. My reply was in response to José Carlos Santos' reply to "Twoflower", and it was intended to give some additional mathematical and historical remarks about this type of function.


Twoflower wrote:

How can we construct function from R to R which, on every open interval, takes every real value?

In 1904 Lebesgue (see p. 90 of [1]) gave a definition of such a function that fits most any notion of constructability I know of. Lebesgue gave a Baire $2$ function that is very simply defined from decimal expansions. For English versions of Lebesgue's function, see [2] and the first two full paragraphs of p. 116 of [3]. Note that Lebesgue's actual example (first full paragraph of p. 116 of [3]) takes, on every open interval, every value in the closed interval $[0,1],$ but there are many ways to easily modify it (for one such way, see the second full paragraph on p. 116 of [3]) so that it takes, on every open interval, every real value. No doubt Lebesgue was aware of this, but the modifications were probably too tangential and trivial for Lebesgue to bother with. For other references and more exotic examples (e.g. a function that takes, in every nonempty perfect set, every real value $2^{\aleph_0}$ many times), see [4]. By the way, the constructions in Halperins' papers (pp. 117-118 in [3] below and reference [3] in [4] below) of a function that takes, in every nonempty perfect set, every real value $2^{\aleph_0}$ times follow immediately from Halperin's earlier construction of a function that takes every real value at least once in every perfect set and the fact that every nonempty perfect set can be written as the union of $2^{\aleph_0}$ many pairwise disjoint perfect sets (a short proof of this fact about perfect sets is given in [5] below). This observation about Halperin's examples is due to Solomon Marcus in 1960 (reference #4 in [4] below).

[1] Henri Lebesgue, Lecons sur l'Intégration et la Recherche des Fonctions Primitives, Gauthier-Villars, 1904.



[2] sci.math -- Non zero Lebesgue measure (Ilias Kastanas, 8 January 1996)


[3] Israel Halperin, Discontinuous functions with the Darboux property, Canadian Mathematical Bulletin 2 (1959), 111-118.



[4] sci.math -- Strange function (Dave L. Renfro, 4 December 2000)


[5] sci.math -- uncountability of the reals (Dave L. Renfro, 1 July 2001)


(added 3 days later)

Searching on-line, I think the earliest usage of the phrase "strongly Darboux" is in the following paper:

Solomon Marcus, From the mean value theorem to a new type of Darboux property, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie (N.S.) 34(82), 1990, 19-23. MR 92b:26008; Zbl 732.26008

The Zbl review doesn't say anything about the phrase and I don't have access to Math. Reviews (only MR Lookup, which is freely available), but I suspect Marcus may be using the term as a variant on the Denjoy-Clarkson property (see [6]).

Ciesielski (e-mail correspondence) believes he made up the term to use in his book.

Regarding Darboux functions in general, if you don't know about the survey paper Bruckner/Ceder [7] (freely available on-line), you should look at it. Although its bibliography is very complete (up to its publication date), the topic of Darboux functions is so widely scattered throughout the older mathematical literature that several omissions are likely to exist. Indeed, in a not very thorough search of my stuff at home I found two omissions that relate to what you're interested in: Wolff [8] and Barrau [9].

There is a (seemingly) relatively recent notion called the Cantor intermediate value property (that I don't know much about) that you might be interested in looking at. Googling the phrase brings up a lot of items freely available on-line, such as Jastrzebski [10]. Finally, Garg [11] (p. 93) uses the phrase weakly Darboux but he doesn't appear to have a phrase for what Ciesielski calls strongly Darboux. [Garg calls $f$ lower Darboux on an interval $I$ if, for each $a < b$ in $I,$ then whenever $f(b) < c < f(a)$ it follows that there exists $x \in I$ such that $f(x) = c.$ For the definition of upper Darboux, $f(b) < c < f(a)$ is replaced with $f(a) < c < f(b).$ Garg calls $f$ weakly Darboux on $I$ when $f$ is lower Darboux on $I$ or $f$ is upper Darboux on $I.$ Note that if $f$ is lower Darboux on $I$ and $f$ is upper Darboux on $I,$ then $f$ is Darboux on $I$.]

[6] sci.math -- Characterization of functions having anti-derivatives (Dave L. Renfro, 21 May 2008)


[7] Andrew Michael Bruckner and Jack Gary Ceder, Darboux continuity, Jahresbericht der Deutschen Mathematiker-Vereinigung 67 (1965), 93-117. MR 32 #4217l; Zbl 144.30003


[8] Julius Wolff, On a function which assumes any value on a non-enumerable set of points in any interval, Proceedings of the Royal Academy of Amsterdam Science 29 (1926), 127-128. JFM 52.0242.03


[9] Johan Antony Barrau, On a function which in any interval assumes any value a non-enumerable number of times, and on a function representing a rectifiable curve which in any interval is non-differentiable a non-enumerable number of times, Proceedings of the Royal Academy of Amsterdam Science 29 (1926), 989-992. JFM 52.0242.05

[10] Jan Jastrzebski, Local characterization of functions having the Cantor intermediate value property, Real Analysis Exchange 24 (1998/99), 223-228. MR 2000b:26006; Zbl 943.26005


[11] Krishna Murari Garg, Theory of Differentiation. A Unified Theory of Differentiation Via New Derivate Theorems and New Derivatives, Canadian Mathematical Society Series of Monographs and Advanced Texts #24, John Wiley and Sons, 1998, xvi + 525 pages. MR 2000c:26002; Zbl 918.26003

  • $\begingroup$ Thanks a lot for your answer, there are plenty of interesting references there. $\endgroup$ Commented May 2, 2012 at 15:54
  • $\begingroup$ Wow, thanks a lot again, you even discussed this with K. Ciesielski. I knew about the paper by Bruckner-Ceder, but I did not know that it is available online; this is a very useful information, too (on the top of other interesting things you wrote.) $\endgroup$ Commented May 4, 2012 at 11:58
  • $\begingroup$ @Martin Sleziak: I've known Ciesielski since the early 1990s, mostly by intermittent email contact, although I have attended approximately 10 conferences that he has attended. FWIW, he, I, and another person decided to leave the rest of our hiking/rock-climbing group at the 22nd Summer Symposium in Real Analysis (June 1998) in Santa Barbara to take on the "difficult course for experienced climbers". (Note: Ciesielski was sufficiently experienced, but I and the other person were not but we managed anyway.) (continued) $\endgroup$ Commented Jun 12, 2012 at 13:41
  • $\begingroup$ @Martin Sleziak: (continued) Regarding strongly Darboux functions, you may want to look at MR 24 #A787. Note that the property that a function map any nonempty perfect set onto the reals is equivalent to the function taking on each real value $2^{\aleph_0}$ times in every nonempty perfect set. Note that the last few words of the review are about the property you are calling "strongly Darboux". $\endgroup$ Commented Jun 12, 2012 at 13:43

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