Differentiability of continuous functions in terms of Haar-smallness

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 superset of A and a continuous map f : {0, 1}N -> C[0, 1] such that f-1[B + h] is Lebesgue's null for all h is an element of 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 is an element of 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. (C) 2020 Elsevier B.V. All rights reserved.