Renormalization theory in Peskin: why he consider massless limit first

In Peskin’s book, he first tackled massless $phi^4$ theory, its renormalization, namely the Callan-Symanzic equation when m=0. And then he put the mass term in later and approach this term ‘s effect by new technics conducted in the session of "renormalization of local operator". I’m just quite confused the reason behind this.
More specificly, I don’t understand the arguments he made on page 409. It seems he claimed that in massless limit the two counterterms $delta_m$ and $delta_Z$ could be treated seperately, but I don’t know what the "continuation" operation means.