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 .
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?)