  Two varieties $Z$ and $\widetilde Z$ are said to be related by extremal
transition if there exists a degeneration from $Z$ to a singular variety
$\overline Z$ and a crepant resolution $\widetilde Z \to \overline Z$. In this
paper we compare the genus-zero Gromov--Witten theory of toric hypersurfaces
related by extremal transitions arising from toric blow-up. We show that the
quantum $D$-module of $\widetilde Z$, after analytic continuation and
restriction of a parameter, recovers the quantum $D$-module of $Z$. The proof
provides a geometric explanation for both the analytic continuation and
restriction parameter appearing in the theorem.
