In this paper, we consider generalizations of two forms of local continuity of maps between topological spaces, with respect to an ideal of subsets either in the domain or in the codomain. In particular, given an ideal I of subsets of a space Y, one-to-one mappings f : X→Y are investigated such that the restriction of f to X0 is continuous for some closed subset X0 of X. Also we investigate continuity of mappings f : X→Y if X is an ideal topological space. Our results provide some co-rollaries for Baire spaces.