  Let $\{f_i:\mathbb{F}_p^i \to \{0,1\}\}$ be a sequence of functions, where
$p$ is a fixed prime and $\mathbb{F}_p$ is the finite field of order $p$. The
limit of the sequence can be syntactically defined using the notion of
ultralimit. Inspired by the Gowers norm, we introduce a metric over limits of
function sequences, and study properties of it. One application of this metric
is that it provides a characterization of affine-invariant parameters of
functions that are constant-query estimable. Using this characterization, we
show that the property of being a function of a constant number of low-degree
polynomials and a constant number of factored polynomials (of arbitrary
degrees) is constant-query testable if it is closed under blowing-up. Examples
of this property include the property of having a constant spectral norm and
degree-structural properties with rank conditions.
