Long and Herrington  have investigated some properties of the strongly ?-continuous functions. In this paper we strengthen some of these properties of Long and Herrington  by studying a new class of functions, called HC-continuous functions.
We introduce a new class of functions called strongly θ-b-continuous function which is a generalization of both strongly θ-precontinuous functionsand strongly θ-semicontinuous functions . Some characterizations and several properties concerning
strongly θ-b-continuous functions are obtained.
In , Açıkgöz et al. introduced and investigated the notions of w-I-continuous and w*-I-continuous functions in ideal topological spaces. In this paper, we investigate their relationships with continuous and θ-continuous functions.