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I was reading the Wikipedia on Clairaut's Theorem (Symmetry of second derivatives) and the article accounts the significant amount of time and failed proofs occurred before the theorem was made fully rigorous by Schwarz in 1873. The article then gives an elementary proof of the theorem using essentially only the Mean Value Theorem, which was proven by Cauchy in 1823.

This makes me wonder why the theorem took 50 additional years after the Mean Value Theorem was proven to be made rigorous, when it does not seem to require many tools, and the ideas do not seem to be too difficult for a mathematician at the time to come across. I was wondering if anyone has any insight to what the field of analysis was like at the time and why there could be such a discrepancy? Could the act of discovering the right conditions (continuity of second partial derivatives) be part of it?

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    $\begingroup$ Related is my answer to Arbitrary Mixed Partial Derivatives. In particular, cited there but worth separately pointing out here, see my 9 June 2007 sci.math post, which gives an ASCII transcription of the historical survey paper A note on the history of mixed partial derivatives by Higgins (1940). Googling the title of this paper will give you some more recent items of possible interest. FYI, I believe the Higgins paper was mostly unknown before I made that sci.math post. $\endgroup$ Commented Dec 8, 2022 at 16:03

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This is because there were many requirements to define what a 'nice' function means. The multivariable case for ensuring continuity is a little more difficult than the single variable case. A function with continuous mixed partials is enough to ensure that mixed partials exist and are equal.

here are the cases highlighted that fail taken from the calculus wiki site linked below

  • Failure of Clairaut's theorem where both mixed partials are defined but not equal
  • Failure of Clairaut's theorem where only one of the mixed partials is defined

Calculus wiki site

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