This paper is concered with the local C-semigroups and local C-coscine functions without the assumption that the image of C is dense in a Banach space X. First, the sufficient and necessary conditions for a local C-semiproup and a C-cosine function to be the restriction of a global C-semigroup and a global C-cosine function to an interval are given,respectively.Secondaly, it is characterized for a closed operator to be the generator of a global C-semigroup and a local C-cosine function,respectively.