  The concept of slice regular function over the real algebra $\mathbb{H}$ of
quaternions is a generalization of the notion of holomorphic function of a
complex variable. Let $\Omega$ be an open subset of $\mathbb{H}$, which
intersects $\mathbb{R}$ and is invariant under rotations of $\mathbb{H}$ around
$\mathbb{R}$. A function $f:\Omega\to\mathbb{H}$ is slice regular if it is of
class $\mathscr{C}^1$ and, for all complex planes $\mathbb{C}_I$ spanned by $1$
and a quaternionic imaginary unit $I$, the restriction $f_I$ of $f$ to
$\Omega_I=\Omega\cap\mathbb{C}_I$ satisfies the Cauchy-Riemann equations
associated to $I$, i.e., $\overline{\partial}_I f_I=0$ on $\Omega_I$, where
$\overline{\partial}_I=\frac{1}{2}\big(\frac{\partial}{\partial\alpha}+I\frac{\partial}{\partial\beta}\big)$.
Given any positive natural number $n$, a function $f:\Omega\to\mathbb{H}$ is
called slice polyanalytic of order $n$ if it is of class $\mathscr{C}^n$ and
$\overline{\partial}_I^{\,n} f_I=0$ on $\Omega_I$ for all $I$. We define global
slice polyanalytic functions of order $n$ as the functions
$f:\Omega\to\mathbb{H}$, which admit a decomposition of the form
$f(x)=\sum_{h=0}^{n-1}\overline{x}^hf_h(x)$ for some slice regular functions
$f_0,\ldots,f_{n-1}$. Global slice polyanalytic functions of any order $n$ are
slice polyanalytic of the same order $n$. The converse is not true: for each
$n\geq2$, we give examples of slice polyanalytic functions of order $n$, which
are not global.
  The aim of this paper is to study the continuity and the differential
regularity of slice regular and global slice polyanalytic functions viewed as
solutions of the slice-by-slice differential equations
$\overline{\partial}_I^{\,n} f_I=0$ on $\Omega_I$ and as solutions of their
global version $\overline{\vartheta}^nf=0$ on $\Omega\setminus\mathbb{R}$. Our
quaternionic results extend to the monogenic case.
