Let $\{Z_n\}$ be a branching process whose offspring distributions vary
with $n$, $\{c_n\}$ a sequence of constants and $W_n=Z_n/c_n$. Write
$F$ for the limit distribution of an a.s. convergent $\{W_n\}$. A necessary
and sufficient condition is given for the support of $F$ to be $[0,\infty]$.
Another necessary and sufficient condition is given for the continuity of $F$
on $(0,\infty)$. Both conditions are formulated in terms of the offspring
variables of the process $\{X_n\}$. The support of $F$ depends on the
positivity or otherwise of $\{P(X_n=i)\}$, while the continuity requires the
divergence of a series derived from $\{P(X_n=i)\}$. Examples of various
types of limit distributions are given.