On preserving continuity in ideal topological spaces

We present some sufficient conditions for continuity of the mapping $f:\langle X,\tau_X^*\rangle \to \langle Y,\tau_Y^*\rangle$, where $\tau_X^*$ and $\tau_Y^*$ are topologies induced by the local function on $X$ and $Y$, resp. under the assumption that the mapping from $\langle X, \tau_X \rangle$ to $\langle Y, \tau_Y \rangle$ is continuous. Further, we consider open and closed functions in this matter, as we state the cases in which the open (or closed) mapping is being preserved through the "idealisation" of both domain and codomain. Through several examples we illustrate that the conditions we considered can not be weakened.