  We propose a self-consistency equation for the $\beta$-function for theories
with a large number of flavours, $N$, that exploits all the available
information in the Wilson-Fisher critical exponent, $\omega$, truncated at a
fixed order in $1/N$. We show that singularities appearing in critical
exponents do not necessarily imply singularities in the $\beta$-function. We
apply our method to (non-)abelian gauge theory, where $\omega$ features a
negative singularity. The singularities in the $\beta$-function and in the
fermion mass anomalous dimension are simultaneously removed providing no hint
for a UV fixed point in the large-$N$ limit.
