  We prove versions of Rad\'o's theorem for polyanalytic functions in one
variable and also on simply connected $\mathbb{C}$-convex domains in
$\mathbb{C}^n$. Let $\Omega\subset \mathbb{C}$ be a bounded, simply connected
domain and let $q\in \mathbb{Z}_+.$ Suppose at least one of the following
conditions holds true: (i) $g\in C^{q}(\Omega).$ (ii) $g\in C^\kappa(\Omega),$
for $\kappa=\min\{1,q-1\},$ such that $g$ is $q$-analytic on $\Omega\setminus
g^{-1}(0)$ and such that Re$g$ (Im$g$ respectively) is a solutions to the
$p'$-Laplace equation ($p''$-Laplace equation respectively) on $\Omega\setminus
g^{-1}(0)$, for some $p',p''>1$. Then $g$ agrees (Lebesgue) a.e.\ with a
function that is $q$-analytic on $\Omega.$ In the process we give a simple
proof of the fact that: If $f\in C^q(\Omega)$ is $q$-analytic on
$\Omega\setminus f^{-1}(0)$ then $f$ is $q$-analytic on $\Omega.$ The
extensions of the results to several complex variables are straightforward
using known techniques.
