A graph $G$ is degree-continuous if the degrees of every two
adjacent vertices of $G$ differ by at most 1.
A finite nonempty set $S$ of integers is convex if $k \in S$ for every
integer $k$ with $\min (S) \leq k \leq \max (S)$. It is shown that
for all integers $r > 0$ and $s \geq 0$ and a convex set $S$ with
$\min (S) = r$ and $\max (S) = r+s$, there
exists a connected degree-continuous graph $G$ with the degree set $S$
and diameter $2s+2$. The minimum order of a degree-continuous
graph with a prescribed degree set is studied.
is shown that for every graph $G$ and convex set $S$ of
positive integers containing the integer 2, there exists a
connected degree-continuous graph $H$ with the degree set $S$ and
$G$ as an induced subgraph if and only if $\max (S)\geq \De (G)$
and $G$ contains no $r-$regular component where $r = \max (S)$.