Summary: We study the map \(\omega : I \times C(I,I) \rightarrow K\) given by \((x,f) \mapsto \omega(x,f)\) that takes a point \(x\) in the unit interval \(I = [0,1]\) and \(f\) a continuous selfmap of the unit interval to the \(\omega\)-limit set \(\omega(x,f)\) that together they generate. We characterize those points \((x,f) \in I \times C(I,I)\) at which \(\omega : I \times C(I,I) \rightarrow K\) is continuous, and show that \(\omega : I \times C(I,I) \rightarrow K\) is in the second class of Baire. We also consider the trajectory map \(\tau : I \times C(I,I) \rightarrow l_{\infty}\) given by \((x,f) \mapsto \tau(x,f) = \{x,f(x),f(f(x)),\ldots\}\) and find that both \(\omega : I \times C(I,I) \rightarrow K\) and \(\tau : I \times C(I,I) \rightarrow l_{\infty}\) are continuous on a residual subset of \(I \times C(I,I)\). We show that the Hausdorff \(s\)-dimensional measure of an \(\omega\)-limit set is typically zero for every \(s > 0\).