2018/10/07 by Kwela, Adam, Wołoszyn, Wojciech Aleksander
#26A27 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1810.03175
One of the classical results concerning differentiability of continuous functions states that the set SD of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space C[0,1]. By a recent result of Banakh et al., a set is Haar-null provided that there is a Borel hull B⊇ A and a continuous map f\colon \0,1\^ℕ→ C[0,1] such that f-1[B+h] is Lebesgue's null for all h∈ C[0,1]. We prove that SD is not Haar-countable (i.e., does not satisfy the above property with "Lebesgue's null" replaced by "countable", or, equivalently, for each copy C of \0,1\^ℕ there is an h∈ C[0,1] such that SD∩ (C+h) is uncountable. Moreover, we use the above notions in further studies of differentiability of continuous functions. Namely, we consider functions differentiable on a set of positive Lebesgue's measure and functions differentiable almost everywhere with respect to Lebesgue's measure. Furthermore, we study multidimensional case, i.e., differentiability of continuous functions defined on [0,1]k. Finally, we pose an open question concerning Takagi's function.