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This issue came to my attention from Math.SE: https://math.stackexchange.com/questions/1015325/is-the-derivative-of-x-on-0-1-always-equal-to-1/1015342#1015342

To summarize, it appears that Mathematica gives erroneous results when computing the numerical value of the derivative of FractionalPart: the command

 Plot[FractionalPart'[x], {x, 0, 2}, WorkingPrecision -> 100,
      MaxRecursion -> 8, PlotPoints -> 100, PlotRange -> {-2, 4}]

generates the following graph:

enter image description here

Verification and/or explanation for this phenomenon is most welcome.

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