  We consider the contact process near an extended surface defect, where the
local control parameter deviates from the bulk one by an amount of
$\lambda(l)-\lambda(\infty) = A l^{-s}$, $l$ being the distance from the
surface. We concentrate on the marginal situation, $s=1/\nu_{\perp}$, where
$\nu_{\perp}$ is the critical exponent of the spatial correlation length, and
study the local critical properties of the one-dimensional model by Monte Carlo
simulations. The system exhibits a rich surface critical behavior. For weaker
local activation rates, $A<A_c$, the phase transition is continuous, having an
order-parameter critical exponent, which varies continuously with $A$. For
stronger local activation rates, $A>A_c$, the phase transition is of mixed
order: the surface order parameter is discontinuous, at the same time the
temporal correlation length diverges algebraically as the critical point is
approached, but with different exponents on the two sides of the transition.
The mixed-order transition regime is analogous to that observed recently at a
multiple junction and can be explained by the same type of scaling theory.
