Let $ A, B \subseteq \mathbb {R} $ be Lebesgue measurable sets such that at least one of them has finite measure. Let $ f $ be the function defined by $f (x) = m ((x + A) \cap B)$ for each $ x \in \mathbb{R} $. Show that $ f $ is continuous.
Hint: Suppose first that $ A $ and $ B $ are intervals and then generalized to arbitrary sets using the regularity of the Lebesgue measure.
Some help please. Thanks.