I was recently looking again at functions like the Cantor staircase, the modified Dirichlet, etc., and something occurred to me. The modified Dirichlet is interesting because it's continuous almost everywhere (i.e. continuous on $[0, 1] \setminus \mathbb{Q}$), but discontinuous on a dense but countable set (i.e. $[0, 1] \cap \mathbb{Q}$). Now $\mathbb{Q}$ is large in the sense of being dense, but is small in most other topological notions (with which I'm familiar), e.g. it's meager, as well as countable, and by countability it has null measure and Hausdorff dimension $0$. That is, the set of discontinuity is quite small in most senses.
Similarly, the set of non-differentiable points of the Cantor staircase is small in the sense of being nowhere-dense, as well as null, though it does have one "size" over $\mathbb{Q}$ in that it has positive Hausdorff dimension. But other than dimension, the set of non-differentiability is fairly small.
My question is this: I'm aware that there are functions like Weierstrass which are always continuous and differentiable nowhere, but what I am not aware of are functions which are continuous everywhere but only differentiable on some "small" set. So my questions are as follows:
(a) Suppose $F \subseteq [0, 1]$ is a set of Hausdorff dimension $s < 1$. Does there exist a map $f: [0, 1] \to \mathbb{R}$ that is differentiable exactly on $F$?
(b) Suppose $F \subseteq [0, 1]$ is a set of Hausdorff dimension $1$ with Lebesgue measure $0$. Does there exist a map $f: [0, 1] \to \mathbb{R}$ that is differentiable exactly on $F$?
(c) Suppose $F \subseteq [0, 1]^{n}$ is a set of Hausdorff dimension $s < n$. Does there exist a map $f: [0, 1]^{n} \to \mathbb{R}^{m}$ that is differentiable exactly on $F$?
(d) Suppose $F \subseteq [0, 1]^{n}$ is a set of Hausdorff dimension $n$ with $n$-dimensional Lebesgue measure $0$. Does there exist a map $f: [0, 1]^{n} \to \mathbb{R}^{m}$ that is differentiable exactly on $F$?
Thanks.
