develop a series of fine methods from the theory of nonlocal equations, comprising boundary Lipschitz bounds, blow-up classification results, continuity implies differentiability results, nonlocal geometric equations and nonlocal obstacle-type problems.

In Section \[HDB\] we construct a useful barrier, that we exploit to rule out the case of boundary Lipschitz singularities for nonlocal minimal graphs.

The proof of Theorem \[NLMS\] is contained in Section \[657699767395535\]. Finally, in Appendix \[APPA\], we point out the classical analogue of Theorem \[FLFL\].

Proof of Theorem \[FLFL\] {#ISsta}
=========================

One important ingredient towards the proof of Theorem \[FLFL\] consists in the construction of a homogeneous solution of a linear fractional equation with a suitable growth from the vertex of a cone. This is indeed a classical topic of research, which also bridges mathematical analysis and probability, see [@MR1936081; @MR2075671; @MR3810469] for specific results on fractional harmonic functions on cones. In our setting, we can reduce to the two-dimensional case (though the higher dimensional case can be treated in a similar way), and, for any $\alpha>0$, we let $$\label{89apq0}
{\mathcal{C}}_\alpha:=\left\{ x=(x_1,x_2)\in{\ensuremath{\mathbb{R}}}^{2}
{\mbox{ s.t. $ x_2>0$ and $x_1+\displaystyle\frac{ x_2 }{\alpha }>0
$}}\right\},$$ and the result that we need is the following one:

\[BAN\] For every $\sigma\in(0,1)$ and every 