  Let $V$ be any shift-invariant subspace of square summable functions. We
prove that if for some $A$ expansive dilation $V$ is $A$-refinable, then the
completeness property is equivalent to several conditions on the local
behaviour at the origin of the spectral function of $V$, among them the origin
is a point of $A^*$-approximate continuity of the spectral function if we
assume this value to be one. We present our results also in the more general
setting of $A$-reducing spaces. We also prove that the origin is a point of
$A^*$-approximate continuity of the Fourier transform of any semiorthogonal
tight frame wavelet if we assume this value to be zero.
