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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.

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    $\begingroup$ 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." $\endgroup$ – Conifold Oct 1 '20 at 0:57
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    $\begingroup$ 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. $\endgroup$ – Conifold Oct 1 '20 at 8:00
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    $\begingroup$ @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. $\endgroup$ – Conifold Oct 1 '20 at 19:49
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    $\begingroup$ 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. $\endgroup$ – Dave L Renfro Oct 2 '20 at 10:49
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    $\begingroup$ @DaveLRenfro Kennedy's book (2002 edition) is accessible on Citeseer. $\endgroup$ – Conifold Oct 2 '20 at 20:51

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