Let $\{Z_n\}$ be a branching process whose offspring distributions vary
with $n$. It is shown that the sequence $\{\max_{i>0}P(Z_n=i)\}$ has a limit.
Denote this limit by $M$. It turns out that $M$ is positive only if the
offspring variables rapidly approach constants.
Let $\{c_n\}$ be a sequence of constants and $W_n=Z_n/c_n$. It will be proven
that $M=0$ is necessary and sufficient for the limit
distribution functions of all convergent $\{W_n\}$
to be continuous on $(0,\infty)$. If
$M>0$ there is, up to an equivalence, only one sequence $\{c_n\}$
such that $\{W_n\}$ has a limit distribution with jump points in $(0,\infty)$.
Necessary and
sufficient conditions for continuity of limit distributions are derived in
terms of
the offspring
distributions of $\{Z_n\}$.