We show that given $n⩾3$, $q⩾1$, and a finite set $\{y_1,\dots ,y_q\}$ in ${\mathbb{R}}^n$ there exists a quasiregular mapping ${\mathbb{R}}^n\to {\mathbb{R}}^n$ omitting exactly points $y_1,\dots ,y_q$.

We prove that the radii of convergence of the solutions of a p-adic differential equation $\mathcal{F}$ over an affinoid domain X of the Berkovich affine line are continuous functions on X that factorize through the retraction of $X\to \mathrm{\Gamma }$ of X onto a finite graph $\mathrm{\Gamma }\subseteq X$. We also prove their super-harmonicity properties. This finiteness result means that the behavior of the radii as functions on X is controlled by a finite family of data.

We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic zero. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorise by the retraction through a locally finite subgraph of the curve.