I have two questions about the so-called Blancmange function (which I'll restrict to having domain $[0,1]$). That is, define:
$$ B:[0,1]\to [0,2],\quad B(x):= \sum_{k = 0} 2^{-k}s\big(2^kx\big)$$
where $s: \mathbb R \to [0,1/2]$ is the sawtooth function:
$$s(x) := \min_{n \in \mathbb Z}\,|x-n| = \min ([x],1-[x]).$$
($[x]$ denoting the fractional part of $x$).
$B$ gives an example of a continuous no-where differentiable function. I was just wondering:
Question 1: What is known about the (Hausdorff/Box Counting) dimension of the graph of $B$? In particular, does the fact that it is self affine lead to an upper bound on the dimension:
$$X = \{(x,B(x))\,|\,x \in [0,1]\} = T_1(X)\cup T_2(X),$$
where $\displaystyle T_1(x,y) = \left(\frac x2,\frac{x+y}2\right)$ and $\displaystyle T_1(x,y) = \left(1- \frac x2,\frac{x+y}2\right)$?
Question 2: Is $B$ of bounded variation?
Plot of $s$:
Plot of $B$:
