  This paper proves continuity of value functions in discounted periodic-review
single-commodity total-cost inventory control problems with
\revision{continuous inventory levels,} fixed ordering costs, possibly bounded
inventory storage capacity, and possibly bounded order sizes for finite and
infinite horizons. In each of these constrained models, the finite and
infinite-horizon value functions are continuous, there exist deterministic
Markov optimal finite-horizon policies, and there exist stationary
deterministic Markov optimal infinite-horizon policies. For models with bounded
inventory storage and unbounded order sizes, this paper also characterizes the
conditions under which $(s_t, S_t)$ policies are optimal in the finite horizon
and an $(s,S)$ policy is optimal in the infinite horizon.
