When a demand function is split, how do we algebraically change function
The Bertrand case you mention is a little special because it induces a discontinuity in the demand function. Suppose that market demand is $D(P)=1-P$, zero marginal cost, and that all consumers buy from the low priced firm (and split equally in the event of a tie).

If the two firms try to collude around a price $p=0.5$ (with quantity $0.25$ each) then each firm's demand curve will look like this:


Mathematically, a firm's demand will be: $Q_i(p_i,p_j)=0$ if $p_i&gt;p_j$; $Q_i(p_i,p_j)=D(p_i)/2$ if $p_i=p_j$; and $Q_i(p_i,p_j)=D(p_i)$ if $p_i&lt;p_j$.



More generally, it is possible for the slope of a firm's residual demand curve to differ from that of the market demand curve. It all depends on how consumers are allocated across the two firms. The relevant material on rationing rules can be found in Section 5.3 of Tirole's The Theory of Industrial Organization.