  Consider a sequence of compactly supported Hamiltonian diffeomorphisms
$\phi_k$ of an exact symplectic manifold, all of which are "graphical" in the
sense that their graphs are identified by a Darboux-Weinstein chart with the
image of a one-form. We show by an elementary argument that if the $\phi_k$
$C^0$-converge to the identity then their Calabi invariants converge to zero.
This generalizes a result of Oh, in which the ambient manifold was the two-disk
and an additional assumption was made on the Hamiltonians generating the
$\phi_k$. We discuss connections to the open problem of whether the Calabi
homomorphism extends to the Hamiltonian homeomorphism group. The proof is based
on a relationship between the Calabi invariant of a $C^0$-small Hamiltonian
diffeomorphism and the generalized phase function of its graph.
