  We have studied the phase transition of the contact process near a multiple
junction of $M$ semi-infinite chains by Monte Carlo simulations. As opposed to
the continuous transitions of the translationally invariant ($M=2$) and
semi-infinite ($M=1$) system, the local order parameter is found to be
discontinuous for $M>2$. Furthermore, the temporal correlation length diverges
algebraically as the critical point is approached, but with different exponents
on the two sides of the transition. In the active phase, the estimate is
compatible with the bulk value, while in the inactive phase it exceeds the bulk
value and increases with $M$. The unusual local critical behavior is explained
by a scaling theory with an irrelevant variable, which becomes dangerous in the
inactive phase. Quenched spatial disorder is found to make the transition
continuous in agreement with earlier renormalization group results.
