Classical theorems from the early 20th century state that any Haar measurable homomorphism between locally compact groups is continuous. In particular, any Lebesgue-measurable homomorphism ϕ : R → R \phi :\mathbb {R} \to \mathbb {R} is of the form ϕ ( x ) = a x \phi (x)=ax for some a ∈ R a \in \mathbb {R} . In this short note, we prove that any Lebesgue measurable function ϕ : R → R \phi :\mathbb {R}\to \mathbb {R} that vanishes under any d + 1 d+1 “difference operators” is a polynomial of degree at most d 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.