Friday, 9 May 2014

real analysis - Points of differentiability of f(x)=sumlimitsn:qn<xcn

Let, {qn}nN be an enumeration of rational numbers. Consider the function f:RR 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 αRQ, we have qn(αδN,α+δN) for 1nN (i.e., say |qnα|>δn for nN where, δn0+ as n) and now if we impose further the 'nice' property:




f(α+δN)f(α)δN=1δNn:qn(α,α+δN)cnλ, (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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