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 talk I plan to prove the following extension: 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, I will prove the continuity of any Haar-measurable polynomial map between locally compact groups, in the sense of Leibman. I will deduce the above result as a direct consequence of a slightly more general theorem about the automatic continuity of cocycles.