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What is the historical motivation of continuous functions? Also, does anyone know who first isolated the idea?

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  • $\begingroup$ According to Wikipedia (and I've read it elsewhere) topology goes back to the ideas of geometria situs (the geometry of place) and analysis situs (the picking apart of place) by Liebniz in the 17th C, unfortunately the article doesn't explain how. $\endgroup$ Commented Nov 21, 2017 at 18:14

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Isolation of the modern concept is generally attributed to Bolzano (1817; translation) and/or Cauchy (1821), with some controversy on their independence: see Grattan-Guinness (1970), Freudenthal (1971). Kline (1972, p. 950) says Bolzano’s motivation was to rigorously prove the fundamental theorem of algebra, and on his way, the intermediate value theorem — stated in his memoir’s title.

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A careful study of the source materials indicates that in fact Cauchy may have beaten Bolzano by a month, and presented a reasonably rigorous definition of continuity in february of 1817, whereas the first written record for Bolzano dates from march 1817; see

Bair, J.; Błaszczyk, P.; Fuentes Guillén, E.; Heinig, P.; Kanovei, V.; Katz, M. "Continuity between Cauchy and Bolzano: Issues of antecedents and priority." British Journal for the History of Mathematics 35 (2020), no. 3, 207-224. https://doi.org/10.1080/26375451.2020.1770015, https://arxiv.org/abs/2005.13259

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