Let $\displaystyle A$ and $\displaystyle B$ be $\displaystyle \lambda$-measurable sets (where $\displaystyle \lambda$ is the Lebesgue measure) such that $\displaystyle 0<\lambda(A),\lambda(B)<\infty$.

Define $\displaystyle f(x)=\lambda((\{x\}+A)\cup B)$. How does one show that $\displaystyle f$ is continuous?