  By classical theorems of Steinhaus and Weil, any Haar measurable homomorphism
between locally compact groups is continuous. In particular, any
Lebesgue-measurable homomorphism $\phi:\mathbb{R} \to \mathbb{R}$ is of the
form $\phi(x)=ax$ for some $a \in \mathbb{R}$. In this short note, we prove
that any Lebesgue measurable function $\phi:\R \to \R$ that vanishes under any
$d+1$ "difference operators" is a polynomial of degree at most $d$. More
generally, we prove the continuity of any Haar measurable polynomial map
between locally compact groups, in the sense of Leibman. We deduce the above
result as a direct consequence of a theorem about the automatic continuity of
cocycles.
