  For all Poincar\'e invariant Lagrangians of the form ${\cal L}\equiv
f(F_{\mu\nu})$, in three Euclidean dimensions, where $f$ is any invariant
function of a non-compact $U(1)$ field strength $F_{\mu\nu}$, we find that the
only continuum limit (described by just such a gauge field) is that of free
field theory: First we approximate a gauge invariant version of Wilson's
renormalization group by neglecting all higher derivative terms $\sim
\partial^nF$ in ${\cal L}$, but allowing for a general non-vanishing anomalous
dimension. Then we prove analytically that the resulting flow equation has only
one acceptable fixed point: the Gaussian fixed point. The possible relevance to
high-$T_c$ superconductivity is briefly discussed.
