  We explore the analytic structure of three-point functions using contour
deformations. This method allows continuing calculations analytically from the
spacelike to the timelike regime. We first elucidate the case of two-point
functions with explicit explanations how to deform the integration contour and
the cuts in the integrand to obtain the known cut structure of the integral.
This is then applied to one-loop three-point integrals. We explicate individual
conditions of the corresponding Landau analysis in terms of contour
deformations. In particular, the emergence and position of singular points in
the complex integration plane are relevant to determine the physical
thresholds. As an exploratory demonstration of this method's numerical
implementation we apply it to a coupled system of functional equations for the
propagator and the three-point vertex of $\phi^3$ theory. We demonstrate that
under generic circumstances the three-point vertex function displays cuts which
can be determined from modified Landau conditions.
