  In the case of finite measures on finite spaces, we state conditions under
which $\phi$-projections are continuously differentiable. When the set on which
one wishes to $\phi$-project is convex, we show that the required assumptions
are implied by easily verifiable conditions. In particular, for input
probability vectors and a rather large class of \phidivergences, we obtain that
$\phi$-projections are continuously differentiable when projecting on a set
defined by linear equalities. The obtained results are applied to the
derivation of the asymptotics of $\phi$-projection estimators (that is, minimum
\phidivergence\, estimators) when projecting on parametric sets of probability
vectors, on sets of probability vectors generated from distributions with
certain moments fixed and on Fr\'echet classes of bivariate probability arrays.
The resulting asymptotics hold whether the element to be $\phi$-projected
belongs to the set on which one wishes to $\phi$-project or not.
