If the Internet connection is lost, control of your heating or air conditioning system is possible via the user interface on the device just as you are used to from a regular thermostat. No Internet connection just means that controlling your heating or air conditioning remotely via the mobile or web app is not possible during this time.
Your apps show zones which cannot access the Internet as zones without remote access. Changes to the settings will not take effect until the connection is reestablished.
When changes are made on the device while the connection to the Internet is not available, the latest changes made on the device are applied in the apps as manual control when the Internet connection is reestablished.