I am trying to understand the above theorem. I get most of the proof but I have trouble understanding why the functions $g_i$ were introduced. I mean why can't we just define $g:X\rightarrow S^n$ by $g(x)=f_i(x)$ if $x\in B_i\cup \bar{V}_i$ instead of $g(x)=g_i(x)$ if $x\in V_i$. What is the need of the $g_i$ functions?
Source: A. R. Pears, Dimension theory of general spaces.