We show that the family of functions in the paper by Sherzad and Mena that give $G_3$ as the inverse limit space, can be expanded to include upper semi-continuous functions whose graphs have a finite number (or even one) of ``short’’ line segments of the form $[x_1,\alpha]\times {a_i}$ and $[\alpha,x_2] \times {b_i}$ where $0 < x_1 < \alpha < x_2 <1$. This is joint work with Sarezh R. Rasul.