Yes, Habyts rules are applied even when the internet connection is down - so your child can't get around the rules by switching off the connection. Their time offline will still tick down and shut down when the time is up.
It does mean that any changes you may make via your Parent Dashboard will not take effect until your child's device goes back online.
On Android use the menu in your notification bar to 'Grant Time' when offline. This needs to be set on your child's actual device. It means that if they get stuck you can free up their device with this function. This is not available on iOS.
However, Habyts works best when it is online - working offline may affect the performance of Habyts in the same way other applications you have installed on your devices can be affected. For example, any settings that the parent changes need an internet connection to pass through, and if the child wants to change from Study Time to Play Time, a connection is needed. We recommend where possible to remain online.