  We derive the discontinuities of banana integrals using the dispersion
relation iteratively. We find a series of identities between the parameterized
discontinuities of banana integrals (p-DOBIs). Similar to elliptic integrals,
these identities enable the reduction of various p-DOBIs to be a linear
combination of some fundamental ones. We present a practical application of
p-DOBIs for the massive three-loop case and establish the expression of
generalized dispersion relation, which enables us to obtain the dispersion
relation representation of arbitrary banana integrals. Moreover, we propose a
hypothesis for generalized dispersion relation and p-DOBIs, which provides a
simple way to calculate the discontinuities and transform dispersion relation
representation to p-DOBIs.
