>>33

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Regarding Peano derivatives, this is easy to find on
the internet. Norton writes: "... the Peano derivative
agrees with the ordinary higher derivatives whenever
the latter is defined, and has the virtue of allowing
us to discuss higher derivatives in the context of a
dense set of discontinuities."

The complete text of Norton's remarks on p. 642 follow,
with minor editing changes to accommodate ASCII format.

Remarks. (1) The Proposition says that g is either not
differentiable at "most" points or infinitely differentiable
at "most" points, according to whether "most" is interpreted
in the sense of category or measure. This is related to the
well-known dichotomy between the Diophantine irrationals
and the Liouville irrationals (those which are not
Diophantine). See [Oxtoby's book] for more on this
interesting topic.

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