The Radio Thermostat CT-30 and Current Innovations CI-300E are the same product, rebranded.
When used standalone, it controls its own schedule like any other thermostat.
When used as a WIFI device, it controls its schedule as updated from the internet portal. There is no 'outside' controller here. The programming is still stored on the thermostat. The (free) Internet portal allows remote changes to current temp or overall schedules and then downloads those schedules back to the thermostat.
When used with Zwave it is by design going to let the Zwave controller handle scheduling and doesn't run any internal schedules to avoid confusion since you now would have 2 different controllers. This is common with Zwave thermostats.
But what if my Zwave network or controller goes down?
If the zwave network goes down it will simply stay at the last temperature. You can manually change it. When the Zwave network is restored, it will pick up the next change the controller sends to it. Since one typically makes moderate set-back adjustments to save money the zwave issue isn't really a major concern and on top of that it doesn't tend to 'go down' unless someone is messing with it or the power fails in which case there isn't any heat or cool running!