  Finite renormalization freedom in locally covariant quantum field theories on
curved spacetime is known to be tightly constrained, under certain standard
hypotheses, to the same terms as in flat spacetime up to finitely many
curvature dependent terms. These hypotheses include, in particular, locality,
covariance, scaling, microlocal regularity and continuous and analytic
dependence on the metric and coupling parameters. The analytic dependence
hypothesis is somewhat unnatural, because it requires that locally covariant
observables (which are simultaneously defined on all spacetimes) depend
continuously on an arbitrary metric, with the dependence strengthened to
analytic on analytic metrics. Moreover the fact that analytic metrics are
globally rigid makes the implementation of this requirement at the level of
local $*$-algebras of observables rather technically cumbersome. We show that
the conditions of locality, covariance, scaling and a naturally strengthened
microlocal spectral condition, are actually sufficient to constrain the allowed
finite renormalizations equally strongly, thus eliminating both the continuity
and the somewhat unnatural analyticity hypotheses. The key step in the proof
uses the Peetre-Slov\'ak theorem on the characterization of (in general
non-linear) differential operators by their locality and regularity properties.
