One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{SD}$ 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 $B\supseteq A$ and a continuous map $f\colon \{0,1\}^\mathbb{N}\to C[0,1]$ such that $f^{-1}[B+h]$ is Lebesgue's null for all $h\in C[0,1]$. We prove that $\mathcal{SD}$ 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 $\{0,1\}^\mathbb{N}$ there is an $h\in C[0,1]$ such that $\mathcal{SD}\cap (C+h)$ 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.

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连续函数在 Haar-smallness 方面的可微性

关于连续函数可微性的经典结果之一指出，某处可微函数（即在某个点可微的函数）的集合 $\mathcal{SD}$ 在空间 $C[0,1] 中是 Haar-null $. 根据 Banakh 等人的最新结果，如果有一个 Borel 外壳 $B\supseteq A$ 和一个连续映射 $f\colon \{0,1\}^\mathbb{N}，那么一个集合是 Haar-null \to C[0,1]$ 使得 $f^{-1}[B+h]$ 是 Lebesgue 对所有 $h\in C[0,1]$ 的空值。我们证明 $\mathcal{SD}$ 不是 Haar-countable（即，不满足上述性质，将“Lebesgue's null”替换为“countable”，或者等效地，对于 $\{0 的每个副本 $C$ ,1\}^\mathbb{N}$ 存在一个 $h\in C[0,1]$ 使得 $\mathcal{SD}\cap (C+h)$ 是不可数的。此外，我们将上述概念用于进一步研究连续函数的可微性。即，我们考虑在一组正 Lebesgue 测度上可微的函数和关于 Lebesgue 测度几乎处处可微的函数。此外，我们研究了多维情况，即定义在 $[0,1]^k$ 上的连续函数的可微性。最后，我们提出了一个关于高木功能的开放性问题。