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Wikipedia cites an earlier result of Cantor as an inspiration but I wonder if there are any previous results of some kind of recursive curve constructions that may have also "inspired" him to come up with that.

Conifold
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    Peano mentions Cantor, Netto and Loria in his paper. He also remarks that the coordinate functions of his curve are continuous nowhere differentiable, so he might have been "motivated" by Weierstrass's 1872 example of such function, which was widely discussed. As Poincare wrote in 1899, "we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions... today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that." – Conifold Oct 01 '20 at 00:57
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    @Conifold A bit of an unrelated question if I may: Offhand, would you know the first name of Netto? I have never seen it in print. In 1862, he proved that an odd perfect number necessarily has at least three prime divisors. I have seen this result quoted at least two or three times, but always attributed to `Netto'. Thanks. – DDS Oct 01 '20 at 04:51
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    @mlchristians The paper in Crelle's journal Peano references is signed "E. Netto", who is Eugen Netto (1846 -1919), a pupil of Weierstrass and Kummer. However, his MacTutor biography does not mention perfect numbers, his bibliography starts with 1870 dissertation, and he was only 16 in 1862. He is the only 19th century Netto Mathematics Genealogy Project lists. Do you have any publication reference for 1862? – Conifold Oct 01 '20 at 07:44
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    Benjamin Peirce proved that an odd perfect number has at least four distinct prime factors already in 1832, so the 1862 date for just three is fishy. – Conifold Oct 01 '20 at 08:00
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    @Conifold That's right (about Peirce) and not too many people seem to know about it. Even J.J. Sylvester who was a friend of Peirce's proved the same result as Peirce, as well as did Servais (independently), over fifty years later in 1888. – DDS Oct 01 '20 at 17:04
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    @Conifold I stand corrected on the 1862 date. According to https://www.emis.de/journals/INTEGERS/papers/d16/d16.pdf (page 3), it was 1863. – DDS Oct 01 '20 at 17:08
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    @Conifold See also, ``Nocco Numbers'' (page 10 or so) http://people.math.harvard.edu/~knill/seminars/perfect/handout.pdf , which makes reference to the 1863 result. Thanks again. – DDS Oct 01 '20 at 17:13
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    @Conifold: so he might have been "motivated" by Weierstrass's 1872 example of such function --- I seriously doubt nowhere differentiability was at the forefront of Peano's thoughts when he was working on this, but instead it was probably somewhat of an afterthought. There were a lot of other pathological functions floating around at the time, based on Henkel's and Cantor's "condensation of singularities method and also found in Dini's work (e.g. his 1878 treatise), and Peano also published several papers involving pathological functions, (continued) – Dave L Renfro Oct 01 '20 at 18:12
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    such as Esempi di funzioni sempre crescenti e discontinue in ogni intervallo AND Sur l'interversion des dérivations partielles AND Sopra alcune curve singolari (English translation in Kennedy's 1973 book) AND Sur la définition de la dérivée (where Peano defines what is now often called the strong derivative), etc. – Dave L Renfro Oct 01 '20 at 18:22
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    @DaveLRenfro I agree, probably not directly. I meant that Weierstrass's example opened the floodgates to the "mass of bizarre functions" and started a trend that Peano and others followed. One pathology gives more confidence for finding more of them. – Conifold Oct 01 '20 at 19:32
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    @mlchristians Ah, so it was Nocco. Dickson's History of the theory of numbers, p. 21 has him as Giovanni Nocco with reference to Alcune teorie su'numeri pari, impari, e perfetti, Lecce, 1863. He does not feature anywhere else, so I am guessing Dickson is everybody else's source. – Conifold Oct 01 '20 at 19:49
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    @Conifold Many thanks!! – DDS Oct 01 '20 at 19:52
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    @Conifold: Weierstrass's example opened the floodgates --- Probably the publication in 1867 of Riemann's 1853 Habilitationsschrift thesis (gives a function continuous on a dense set that is discontinuous on another dense set) followed by Hankel's 1870 memoir (has a lot of constructions of pathological functions based on ideas introduced by Riemann; by the way, G. Cantor published a review of Hankel's memoir in 1871). Among other things, in this memoir Hankel (continued) – Dave L Renfro Oct 02 '20 at 09:57
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    provided the first (publicly presented, in light of Bolzano's function) example of a continuous (or otherwise) function that is differentiable on a dense set and not differentiable on another dense set, a distinction from nowhere differentiable that many people at that time did not make (and which was probably seen by most everyone as every bit as pathological as nowhere differentiable), although people like Darboux and Dini, and du Bois-Reymond certainly knew of the distinction then. (continued) – Dave L Renfro Oct 02 '20 at 10:07
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    Indeed, Darboux almost gave the first (publicly announced) continuous nowhere differentiable function (and at the time he was not yet aware of the Weierstrass example), when he presented examples at French Mathematical Society meetings on 19 March 1873 and 28 January 1874. However, by the early to mid 1880s I think it was becoming more generally known that such a function exists, and the Weierstrass function (via its appearance in the 1875 paper by du Bois-Reymond) started to be mentioned in an occasional "analysis treatise". (continued) – Dave L Renfro Oct 02 '20 at 10:23
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    I think it wasn't until the 1890s that the Weierstrass function started making regular appearances in analysis treatises and textbooks, and thereby started becoming more generally known to non-experts. As for Peano's motivation, I think Jordan's late 1880s work on arc length, Scheeffer's work, Cantor connectedness (see this 16 April 2007 sci.math post), Netto's work, etc. made the possibility of Peano's curve as something that was "in the air" at the time. – Dave L Renfro Oct 02 '20 at 10:49
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    @DaveLRenfro Thank you, this is very informative. I think this is more than enough for an answer if you get a chance. – Conifold Oct 02 '20 at 11:24
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    @Conifold: I'll try to assemble these together, along with more relevant context that I can think of, when I get a chance (might take two or three days). However, I don't really know more precisely an answer to the question. I held off because I assumed it would be in Kennedy's 1973 biography of Peano (a book I don't have, although I photocopied some of Peano's translated papers from it long ago, probably in the mid 1990s) and someone who has this would be able to say a bit more precisely. Someone with access to Kennedy's book want to look into this? – Dave L Renfro Oct 02 '20 at 13:48
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    @DaveLRenfro Kennedy's book (2002 edition) is accessible on Citeseer. – Conifold Oct 02 '20 at 20:51

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