if f is continuous then every preimage of lebesgue measurable is lebesgue measurable

Question

We know that if $f: A\to B$ is continuous and $E\subseteq B$ is a measurable Borel set, then $f^{-1}(E)\subseteq A$ is Borel measurable.

My question is that, If $E\subseteq B$ is Lebesgue measurable and $A=B=\mathbb{R}$ then how can prove that $f^{-1}(E)$ is Lebesgue measurable?