  We study the problem of testing whether a function $f: \mathbb{R}^n \to
\mathbb{R}$ is a polynomial of degree at most $d$ in the
\emph{distribution-free} testing model. Here, the distance between functions is
measured with respect to an unknown distribution $\mathcal{D}$ over
$\mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we
do not assume that $\mathcal{D}$ has finite support.
  We design a tester that given query access to $f$, and sample access to
$\mathcal{D}$, makes $(d/\varepsilon)^{O(1)}$ many queries to $f$, accepts with
probability $1$ if $f$ is a polynomial of degree $d$, and rejects with
probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with
$f$ on a set of mass at least $\varepsilon$ with respect to $\mathcal{D}$. Our
result also holds under mild assumptions when we receive only a polynomial
number of bits of precision for each query to $f$, or when $f$ can only be
queried on rational points representable using a logarithmic number of bits.
Along the way, we prove a new stability theorem for multivariate polynomials
that may be of independent interest.
