$\vartheta\in{\ensuremath{\mathbb{R}}}$ there exist $\alpha>0$, $\beta\in(0,2\sigma)\setminus\{ \vartheta\}$ and $\bar{v}:S^1\to [0,+\infty)$ with $\| \bar{v}\|_{L^\infty(S^{1})}=1$ such that the function $$\label{eca-a} {\ensuremath{\mathbb{R}}}^2\ni x\mapsto v(x):=|x|^\beta\,\bar{v}\left( \frac{x}{|x|}\right)$$ satisfies $$\label{eca} \begin{cases}
(-\Delta)^\sigma v = 0 & {\mbox{ in }}{\mathcal{C}}_\alpha,\\
v=0 & {\mbox{ in }}{\ensuremath{\mathbb{R}}}^2\setminus{\mathcal{C}}_\alpha.\end{cases}$$

Let $\vartheta\in{\ensuremath{\mathbb{R}}}$. By Theorem 3.2 in [@MR2075671], for every $\alpha>0$ there exist $\beta(\alpha)\in(0,2\sigma)$ and $\bar{v}_\alpha:S^1\to [0,+\infty)$ with $\| \bar{v}_\alpha\|_{L^\infty(S^{1})}=1$ such that the function $${\ensuremath{\mathbb{R}}}^n\ni x\mapsto v_\alpha(x):=|x|^\beta\,\bar{v}_\alpha\left( \frac{x}{|x|}\right)$$ satisfies $$\begin{cases}
(-\Delta)^\sigma v_\alpha = 0 & {\mbox{ in }} {\mathcal{C}}_\alpha,\\
v_\alpha=0 & {\mbox{ in }}{\ensuremath{\mathbb{R}}}^2\setminus{\mathcal{C}}_\alpha.\end{cases}$$ Let us focus on the case $\alpha=1$. If $\beta(1)\ne\vartheta$, then the claims of Lemma \[BAN\] are satisfied by choosing $\alpha:=1$, $\beta:=\beta(1)$ and $\bar{v}:=\bar{v}_1$.

If instead $\beta(1)=\vartheta$, we exploit Lemma 3.3 in [@MR2075671]. Namely, since ${\mathcal{C}}_{1/2}\supset {\mathcal{C}}_1$, we deduce from Lemma 3.3 in [@MR2075671] that $\beta(1/2)<
\beta(1)=\vartheta$, and thus the claims of Lemma \[BAN\] are satisfied in this case by choosing $\alpha:=1/2$, $\beta:=\beta(1/2)$ and $\bar{v}:=\bar{v}_{1/2}$.

Exploiting Lemma \[BAN\], we will obtain that the boundary derivatives of $\sigma$-harmonic functions have maximal span. For this, given $k\in{\ensuremath{\mathbb{N}}}$, we define by ${\mathcal{H}}_k$ the set of all functions $u\in C({\ensuremath{\mathbb{R}}}^n)$ for which there exists $r>0$ such that $$\label{KK} \begin{cases}
(-\Delta)^\sigma u=0 & {\mbox{ in }} B_r\cap \{x_n>0\},\\
u=0 & {\mbox{ in }} \{x_n<0\},\end{cases}$$ and for 