A BLYNK_WRITE() function is called whenever a corresponding widget is touched/triggered… if the App is offline, then obviously this cannot happen (technically the API call or a write/sync command combo will work… but that requires the server for sure to be connected to the device)
But depending on how you write your code, the previous value is already assigned to setpoint, or if the device itself is rebooting, can be recalled from the server with a Blynk.syncVirtual(V16) command.
Then your device is totally dependent on how you wrote your code (able to function when disconnected from server or not), and whether or not it is still running with last values retained in the variable or not.
Depends on the widget… the Time Input widget is just a simple data collection widget that sends both start and stop times to your device as soon as you enter them… then it is up to your code to remember and process them “on time”… thus needs to be running and probably connected to the server, unless you have implemented connection management code as per link above.
However, I think the Timer Widget should process accordingly on time IF the server is connected to the device, even if App not active… You should be able to test this on your own with a simple sketch.