Let X. Y be compact Hausdorff spaces, and E, F be Banach spaces. A linear map T¡GC (X¡AE)¡÷C (Y¡AF) is separating if ¡üTf(y)¡ü¡üTg(y)¡ü¡×0 whenever ¡üf(x)¡ü¡üg(x)¡ü¡×0, for every x belonging to X, y belonging to Y. Gau, Jeang and Wong prove that a biseparating linear bijection T is a weighted composition oprator Tf¡×hf¡³£p where h is a function from Y into the set of inveritable linear operators from E onto F and £p is a homeomorphism from Y onto X. In this thesis, we extend this result to the case that continuous functions are defined to a locally compact Hausdorff space, which is either £m-compact or first countable. Moreover, we give a short proof of a recent result of Mrcun. Finally, we give an alternative approach to an Araujo's result concerning biseparating maps of smooth functions appeared in Adv. Math.