Continuous and preserves measurability $\implies$ preserves null sets.

Prove this by contrapositive:

Suppose that $f$ doesn't preserve null sets. Then there is a set $f(N)$ that does not have zero measure. That is, $f(N)$ must have a non-measurable subset $A$.

Now, consider $f^{-1}(A) \cap N$, which must be measure zero, by the completeness of Lebesgue measure.