Using the homotopy invariance property of the degree and a newly introduced concept of the interior-point-epsilon-exceptional family for continuous functions, we prove an alternative theorem concerning the existence of a certain interior-point of a continuous complementarity problem. Based on this result, we develop several sufficient conditions to assure some desirable properties (nonemptyness, boundedness, and upper-semicontinuity) of a multivalued mapping associated with continuous (nonmonotone) complementarity problems corresponding to semimonotone, P(tau, alpha, beta) -, quasi- P*-, and exceptionally regular maps. The results proved in this paper generalize well-known results on the existence of central paths in continuous P-0 complementarity problems.