  It is shown that the renormalization group turns to be a symmetry group in a
theory initially formulated in a space of scale-dependent functions, i.e, those
depending on both the position $x$ and the resolution $a$. Such theory, earlier
described in {\em Phys.Rev.D} 81(2010)125003, 88(2013)025015, is finite by
construction. The space of scale-dependent functions $\{ \phi_a(x) \}$ is more
relevant to physical reality than the space of square-integrable functions
$\mathrm{L}^2(R^d)$, because, due to the Heisenberg uncertainty principle, what
is really measured in any experiment is always defined in a region rather than
point. The effective action $\Gamma_{(A)}$ of our theory turns to be
complementary to the exact renormalization group effective action. The role of
the regulator is played by the basic wavelet -- an "aperture function" of a
measuring device used to produce the snapshot of a field $\phi$ at the point
$x$ with the resolution $a$. The standard RG results for $\phi^4$ model are
reproduced.
