As asked in the title:

Are there any written sources (from the 19th century) explicitly stating the belief that any function satisfying the intermediate value property is continuous?

(I do not believe it makes sense to search for earlier sources, since the notion of continuity itself was not made rigorous until the 19th century. This question originated in an answer I gave at Math.Stackexchange. What follows borrows heavily from that answer.)

If I is an interval, and f : I → ℝ, we say that f has the intermediate value property if and only if whenever a≠b are points of I, if c is between f(a) and f(b), then there is a d between a and b with f(d)=c.

Bolzano published in 1817 his paper Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation). There, he proves that continuous functions satisfy the intermediate value property. As he indicates in the paper, the proposition was widely believed to be true, and several "geometric" arguments had been given trying to justify it.

On the other hand, now we know that the intermediate value property is far weaker than continuity. A nice survey containing detailed examples of functions that are discontinuous and yet have the intermediate value property is

I. Halperin, Discontinuous functions with the Darboux property, Can. Math. Bull., 2 (2), (May 1959), 111-118.

In Halperin's paper we find the amusing quote

Until the work of Darboux in 1875 some mathematicians believed that [the intermediate value] property actually implied continuity of f(x).

This claim is repeated in (many) other places. For example, here one reads

In the 19th century some mathematicians believed that [the intermediate value] property is equivalent to continuity.

This is very similar to what we find in A. Bruckner, Differentiation of real functions, AMS, 1994. In page 5 we read

This property was believed, by some 19th century mathematicians, to be equivalent to the property of continuity.


Historically, this intermediate value property has been suggested as a definition for continuity of real-valued functions [citation needed].

I have been unable to find a direct source expressing this belief. That this was indeed the case is perhaps supported by the following two quotes from Gaston Darboux's Mémoire sur les fonctions discontinues, Ann. Sci. Scuola Norm. Sup., 4, (1875), 161–248. First, on pp. 58-59 we read:

Au risque d'être trop long, j'ai tenu avant tout, sans y réussir peutêtre, à être rigoureux. Bien des points, qu'on regarderait à bon droit comme évidents ou que l'on accorderait dans les applications de la science aux fonctions usuelles, doivent être soumis à une critique rigoureuse dans l'exposé des propositions relatives aux fonctions les plus générales. Par exemple, on verra qu'il existe des fonctions continues qui ne sont ni croissantes ni décroissantes dans aucun intervalle, qu'il y a des fonctions discontinues qui ne peuvent varier d'une valeur à une autre sans passer par toutes les valeurs intermédiaires.

Darboux's paper proves that derivatives have the intermediate value property, and that there are discontinuous derivatives, thus first verifying that the two notions are not equivalent. (For this reason, the intermediate value property is sometimes called the Darboux property or, even, one says that a function with this property is Darboux continuous.)

The proof that derivatives have the intermediate value property starts on page 109, where we read:

En partant de la remarque précédente, nous allons montrer qu'il existe des fonctions discontinues qui jouissent d'une propriété que l'on regarde quelquefois comme le caractère distinctif des fonctions continues, celle de ne pouvoir varier d'une valeur à une autre sans passer par toutes les valeurs intermediaires.

Wikipedia mentions the following:

Before the formal definition of continuity was given, the intermediate value property was given as part of the definition of a continuous function. Proponents include Louis Arbogast, who assumed the functions to have no jumps, satisfy the intermediate value property and have increments whose sizes corresponded to the sizes of the increments of the variable.

The article cites this site, though I have been unable to verify this from Arbogast's writings (or from the linked site). Indeed, Arbogast seems to have a notion of function that is significantly more restrictive that our modern notion of continuity, and therefore the intermediate value theorem holds there. I do not see that he directly addresses the intermediate value property, or indicates that it implies continuity. (Given his understanding of what a function is, I am not even certain that this would have been meaningful.)

Finally, let me ask:

If it is not actually the case that the belief in the equivalence of these two notions was explicitly stated in the literature, where does the false claim originate? (Is it in Halperin's paper?)

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    $\begingroup$ I've been recently made aware of MR0165049 (29 #2340). Valabrega, Elda Gibellato. Il teorema di esistenza degli zeri delle funzioni continue nell'analisi moderna. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 98 1963/1964 437–444. The paper seems to concern itself, at least in part, with precisely this question. I will expand this into an answer once I've read the paper carefully and confirmed its relevance. $\endgroup$ – Andrés E. Caicedo Oct 21 '15 at 19:39

One answer to your question could be that the separation actually came pretty late. Wikipedia claims that "Earlier authors held the result to be intuitively obvious, and requiring no proof.", so until continuity was formalized by Bolzano and Cauchy, I believe it doesn't make sense to find evidence. So we need to look for people who read Bolzano or Cauchy, and believed that intermediate value property is equivalent to continuity.

As you already stated in your question, Darboux showed in 1875 you could verify the theorem without being continuous. This leaves a small window - 1817-1875 - to find for published absurdities.

And here comes Darboux himself.

La propriété précédente a souvent étée prise pour la définition des fonctions continues

which translates :

The aforementioned proposition was often confused with the definition of continuous fonction

So this answers your second question: if no prior evidence can be found, Darboux himself made the claim that the error was widespread before his own work.

In the introduction of the same memoire, Darboux states that M. Hankel's work of 1870 regarding Riemann's memoire where not beyond any reproach, but it's not clear whether he is talking about the existency of a derivative to all functions or the intermediate value theorem at this point. I believe one willing to find evidence of a confusion could look into M. Hankel's work, but I couldn't find the paper which Darboux describes.

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    $\begingroup$ Yes, thank you. I agree that only articles after Bolzano's and prior to Darboux's seem relevant. I also indicated in the question that Darboux suggests that the error was "common" (the two quotes I chose meant to illustrate this). I haven't seen Hankel's article either; I'll see if I can get a copy in the next few days. $\endgroup$ – Andrés E. Caicedo Nov 2 '14 at 21:38

This is not really an answer, but is too long to fit in a comment. The question presupposes that according to 19th-century definitions, it is false that the intermediate value property implies continuity. It's far from clear to me that this was the case.

There are many possible ways of formulating the definition of the function. Three examples would be to define the notion as a formula, to use point-set notions, or to proceed as in modern smooth infinitesimal analysis (SIA). As far as I can tell from the WP article "History of the function concept," the point-set version didn't become fully developed and universally accepted until well into the 20th century.

If we have a counterexample to the claim, then for every real $y$, we have a set $S_x$ of $x$ values equinumerous to the rationals, with all the $S_x$ disjoint and lying in a finite interval. This is equivalent to a proof that $\mathbb{R}$ is equinumerous to $\mathbb{R}\times\mathbb{Q}$. This requires at least the following:

(1) We accept the existence of functions that are discontinuous everywhere.

(2) We accept the Cantorian analysis of infinities.

These are both significant philosophical choices, not inevitable truths. #1 is false in SIA, for example. #2 was highly controversial at the end of the 19th century.

So I think a better way to ask the question might be more like this: at what point in the 19th or 20th century did there develop a sufficient consensus about definitions and philosophy to make it false, according to the standard choices, that the intermediate value property implies continuity?

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Frankly, I don't understand how anybody in 19th century could have thought that intermediate value property implies continuity. Take $y(x)=0$ if $x=0$ and $y=sin(1/x)$ otherwise and you have your counterexample.

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    $\begingroup$ I suspect that functions defined by case analysis on the reals where taken into consideration rather late. Anyone know details? $\endgroup$ – Michael Bächtold Jun 14 '19 at 8:04

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