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The concept of Lipschitz continuous mappings is probably at the present time the most important mathematical concept associated with Lipschitz's name. These mappings play an important role in the theory of differential equations and they also appear in many other branches of mathematics.

What was R. Lipschitz' original motivation for considering these mappings?

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    $\begingroup$ "one of the most important mathematical achievements of Rudolf Lipschitz" I would probably phrase this as "at the present time one of the most important mathematical concepts associated with Lipschitz's name". $\endgroup$ – Dave L Renfro Jan 23 '18 at 16:25
  • $\begingroup$ @DaveLRenfro I agree. Please feel free to edit and change it. $\endgroup$ – Christian Jan 23 '18 at 16:32
  • $\begingroup$ OK, I made a few quick editing touch-ups to it. Feel free to change things if you don't like something I did. $\endgroup$ – Dave L Renfro Jan 23 '18 at 17:09
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    $\begingroup$ Short discussion in Janhke's A History of Analysis $\endgroup$ – Michael E2 Jan 23 '18 at 19:02
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It is existence and uniqueness question for ordinary differential equations. When I was a student (in 1970s) Lipschitz functions were not omnipresent in Analysis. The only context where this name appeared (in undergraduate curriculum) was existence and uniqueness theorem for ODE. And this was apparently his original motivation, as the reference in the comment of @Michael E2 shows. Nowadays this condition has more wide use because of the development of non-smooth analysis in the recent decades.

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