of the solution in the vicinity of the boundary.

As a first result, we will show here that, roughly speaking, the function in  can be [*arbitrarily prescribed, up to an arbitrarily small error*]{}. That is, one can construct solutions of linear fractional equations whose fractional boundary derivative behaves in an essentially arbitrary way.

Then, we turn our attention to the boundary property of [*nonlocal minimal graphs*]{}, i.e. minimizers of the fractional perimeter functional which possess a graphical structure. In this case, we show that the boundary properties are subject to severe geometric constraints, in sharp contrast with the case of fractional equations.

First of all, the continuity properties of nonlocal minimal graphs are very different from those of the solutions of fractional equations, since we have established in [@MR3516886; @MR3596708] that nonlocal minimal graphs are not necessarily continuous at the boundary. In addition, the boundary discontinuity of nonlocal minimal graphs in the plane happens to be a “generic” situation, as we have recently proved in [@2019arXiv190405393D].

The focus of this article is on the three-dimensional setting, i.e. the case in which the graph is embedded in ${\ensuremath{\mathbb{R}}}^3$. In this situation, the graph can be continuous at a given point, but 