Let, {qn}n∈N be an enumeration of rational numbers. Consider the function f:R→R given by, f(x)=∑n:qn<xcn
where, ∞∑n=1cn is an absolutely convergent positive series. The function is clearly monotone increasing, discontinuous at rationals (with jump exactly cn at x=qn) and continuous at irrationals.
1. I wish to ask about the points of differentiability of f?
Since f is monotone it should be differentiable a.e. but how do we identify these points of differentiability? (as in a way of representing this set in a compact way)
Intuitively, it seems they should be related to the particular enumeration of the rationals {qn} at hand. For example if we have an enumeration such that for α∈R∖Q, we have qn∉(α−δN,α+δN) for 1≤n≤N (i.e., say |qn−α|>δn for n∈N where, δn↓0+ as n→∞) and now if we impose further the 'nice' property:
f(α+δN)−f(α)δN=1δN∑n:qn∈(α,α+δN)cn→λ, (as N→∞) and similarly one for the left derivative, we have f′(α)=λ.
So, intuitively I can see how to choose an enumeration that makes the derivative equal λ at x=α (or blows up at α, i.e., λ=+∞).
To clarify what I am asking: Given an enumeration of rationals, how do we come up with relevant definitions/concepts relating to said enumeration, which helps us identify which α's we should expect to be a point of differentiablity.
2. Is there a way to estimate the derivative at these points?
Has these questions been addressed/answered in literature before? I'd love it if I could get some reference in this matter. Thanks! :-)
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