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From beginnings of topology, it was clear that the closed interval $\left[a,b \right]$ of the real line had a certain property that was crucial for proving such theorems as the maximum value theorem and the uniform continuity theorem.(Sect-26, Munkres)

What was actually the difficulty faced by mathematicians in precisy-fying the intuitive idea of continuity?

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    $\begingroup$ See A pedagogical history of compactness by Manya Raman-Sundstrom (2015) AND La genèse du théorème de recouvrement de Borel by Bernard Maurey and Jean-Pierre Tacchi (2005) AND The Borel theorem and its generalizations by Hildebrandt (1926). $\endgroup$ Sep 2, 2022 at 17:14
  • $\begingroup$ The difficulty was that what was precisified was not the intuitive idea of continuity, as reactions to Weierstrass's nowhere differentiable continuous function and Peano's square filling curve showed. Hermite "turned away with fright and horror from this lamentable evil", Poincare said that "logic sometimes makes monsters" and complained that "one invents them purposely to show up defects in the reasoning of our fathers", and only that. What eventually happened was a triumph of technical utility over intuition, and it took time for that utility to become apparent in various contexts. $\endgroup$
    – Conifold
    Sep 2, 2022 at 19:03

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