Idealization of Ganster-Reilly decomposition theorems

In 1990, Ganster and Reilly proved that a function is continuous if and only if it is precontinuous and LC-continuous. In this paper we extend their decomposition of continuity in terms of ideals. We show that a function $f \colon (X,\tau,{\cal I}) \to (Y,\sigma)$ is continuous if and only if it is pre-I-continuous and I-LC-continuous. We also provide a decomposition of I-continuity.