= Solution
The bound $0\leq d_n\leq2^{-n-1}$ proves <uniform convergence> of the series by the <Weierstrass M-test>. Each summand is continuous and satisfies $d_n(x+1)=d_n(x)$, so its sum $t$, the <Takagi function>, is continuous and has period one.
For $n\geq m$, both $2^nu_m$ and $2^nv_m$ are integers, giving $d_n(u_m)=d_n(v_m)=0$. Therefore
$$
S_m:=\frac{t(v_m)-t(u_m)}{v_m-u_m}
=\sum_{n=0}^{m-1}\frac{d_n(v_m)-d_n(u_m)}{v_m-u_m}.
$$
The corners of $d_n$ occur on the grid $2^{-n-1}\mathbb Z$. When $n<m$, that grid is contained in $2^{-m}\mathbb Z$, so the interior of $[u_m,v_m]$ contains no corner of $d_n$. Its slope on this interval is consequently $+1$ or $-1$.
The interval $[u_{m+1},v_{m+1}]$ is a half of $[u_m,v_m]$. All the old slopes persist on this half, and the new summand contributes one more slope in $\{\pm1\}$. Thus
$$
\boxed{S_{m+1}-S_m\in\{\pm1\}.}
$$
If $t$ had a finite <derivative> $L$ at $x_0$, write $t(y)=t(x_0)+L(y-x_0)+r(y)$ with $r(y)=o(|y-x_0|)$. Since $u_m\leq x_0<v_m$ and both endpoints tend to $x_0$,
$$
|S_m-L|\leq\frac{|r(v_m)|+|r(u_m)|}{v_m-u_m}\longrightarrow0.
$$
Indeed $|v_m-x_0|+|u_m-x_0|=v_m-u_m$; when $u_m=x_0$, its remainder is zero. This is the <secants straddling a differentiability point> argument, including dyadic $x_0$. But the displayed increments prevent $S_m$ from converging. Hence $\boxed{t\text{ is nowhere differentiable}}$, using periodicity to cover all real $x_0$.
For the <distributional derivative>, set $r_n(x)=s(2^nx)$, using $s$ from part (iii). The change of variable in the <distributional derivative> pairing gives $d_n'=r_n$; these are the <Rademacher functions>. <Uniform convergence> implies <weak convergence of distributions>, and the <continuity of distributional differentiation> allows termwise differentiation in that topology. Explicitly,
$$
\boxed{t'=\sum_{n=0}^{\infty}r_n\quad\text{in }\mathcal D'(\mathbb R),\qquad
\langle t',\phi\rangle=-\int_{\mathbb R}t(x)\phi'(x)\,dx.}
$$
This series assertion is a <distributional convergence> statement. Its remainder satisfies the concrete estimate
$$
\left|\left\langle t'-\sum_{n=0}^{N}r_n,\phi\right\rangle\right|
\leq2^{-N-1}\|\phi'\|_{L^1}\longrightarrow0.
$$
It supplies a well-defined <distributional derivative> even though the ordinary <derivative> exists nowhere.
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