I am studying space filling curves and I am using the Hans Sagan book.
I am trying to understand the nowhere differentiability of the Hilbert curve presented in this book but it does not seem to make much sense to me.
So does anyone have an analytical proof that the Hilbert curve is nowhere differentiable?
The Hilbert curve is defined as the mapping $f_h: I \to \mathcal{Q}$ where I in the unit interval in $\mathbb{R}$ and $\mathcal{Q}$ is the unit square.
$$f_h=(0_4.q_1q_2q_3...)=\sum^\infty_{j=1} (1/2^j)(-1)^{e_{0j}} sgn(q_j)\left(\begin{array}{cc} (1-d_j)g_j-1 \\ 1-d_jq_j \end{array} \right)$$
I'm surprised there's a simple formula like that that works for every $t$. If a given $t$ has two different quaternary representations does the formula give the same answer for both representations? (If not we have to choose one, and you left out the statement of which one we use.)
Anyway, it's clear that $f$ is nowhere-differentiable if you look at it right. Fix $t\in[0,1)$. For every $n$ there exists $j$ with $$t\in I_n=[j4^{-n},(j+1)4^{-n}].$$ If $n$ is large enough then $$I_n'=[(j+1)4^{-n},(j+2)4^{-n}]\subset[0,1].$$Now $I_n$ is mapped onto a square $Q_n$ of side length $2^{-n}$, and $I_n'$ is mapped onto a similar square $Q_n'$. Since $Q_n$ and $Q_n'$ are disjoint except for sharing an edge, if $s_n\in I_n'$ is chosen to maximize $|f(t)-f(s_n)|$ then $$|f(t)-f(s_n)|>2^{-n}.$$But $|t-s_n|\le4^{-n+1}$. So $s_n\to t$ but $$\frac{|f(t)-f(s_n)|}{|t-s_n|}\to\infty,$$hence $f$ is not differentiable at $t$.