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I was conversing with gpt and it made a claim that the set of points where a monotonic function $f$ ( defined on [0,1] ) is not differentiable is a set of first category.

While asking for a proof it spit out two other problems

  1. The upper dini derivative is upper semi-continous for monotonic function.

  2. The set $\{x : Df(x) \geq a\} $is closed for monotonic function

Where the upper dini derivative is $Df(x) = \lim_{ k\rightarrow 0+}\sup_{0<h<k} \frac{f(x+h) - f(x) } {h} $

The sources referred to by gpt was book Saks - Theory of Integration, but i could not find ( and understand the writing style) the proof in there.

Any help is appreciated for all the above three queries.

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  • $\begingroup$ @Eric Towers: The question/answers you cited deal with "almost all" in the sense of Lebesgue measure, but the OP is asking about "almost all" in the sense of Baire category. This 4 November 2000 sci.math post gives a survey (up to what was known by 2000) of non-differentiability of monotone functions that includes "almost all" in other ways besides Lebesgue measure. Briefly, a continuous strictly increasing function can be non-differentiable on the complement of a first category set, (continued) $\endgroup$ Commented Nov 15 at 20:52
  • $\begingroup$ and that set of non-differentiability can simultaneously have Hausdorff dimension $1$ (although it always has Lebesgue measure zero). Incidentally, Zamfirescu (1981 & 1984) proved that MOST nondecreasing continuous functions on [0,1] are not differentiable at MOST points, when “MOST” is used the Baire category sense in both places. $\endgroup$ Commented Nov 15 at 21:06

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