  A class of continuum models with a critical end point is considered whose
Hamiltonian ${\mathcal{H}}[\phi,\psi]$ involves two densities: a primary
order-parameter field, $\phi$, and a secondary (noncritical) one, $\psi$.
Field-theoretic methods (renormalization group results in conjunction with
functional methods) are used to give a systematic derivation of singularities
occurring at critical end points. Specifically, the thermal singularity
$\sim|{t}|^{2-\alpha}$ of the first-order line on which the disordered or
ordered phase coexists with the noncritical spectator phase, and the
coexistence singularity $\sim |{t}|^{1-\alpha}$ or $\sim|{t}|^{\beta}$ of the
secondary density $<\psi>$ are derived. It is clarified how the renormalization
group (RG) scenario found in position-space RG calculations, in which the
critical end point and the critical line are mapped onto two separate fixed
points ${\mathcal P}_{\mathrm{CEP}}^*$ and ${\mathcal P}_{\lambda}^*$
translates into field theory. The critical RG eigenexponents of ${\mathcal
P}_{\mathrm{CEP}}^*$ and ${\mathcal P}_{\lambda}^*$ are shown to match.
${\mathcal P}_{\mathrm{CEP}}^*$ is demonstrated to have a discontinuity
eigenperturbation (with eigenvalue $y=d$), tangent to the unstable trajectory
that emanates from ${\mathcal P}_{\mathrm{CEP}}^*$ and leads to ${\mathcal
P}_{\lambda}^*$. The nature and origin of this eigenperturbation as well as the
role redundant operators play are elucidated. The results validate that the
critical behavior at the end point is the same as on the critical line.
