Continue operation when app did enter background on iOS

That's not how you do this. Any code that you want to run in the background must be wrapped properly. Something like this:

- (void)someMethodToKeepRunningInBackground {
    UIBackgroundTaskIdentifier taskId = [[UIApplication sharedApplication] beginBackgroundTaskWithExpirationHandler:^(void) {
        // Uh-oh - we took too long. Stop task.
    }];

    // Perform task here        

    if (taskId != UIBackgroundTaskInvalid) {
        [[UIApplication sharedApplication] endBackgroundTask:taskId];
    }
}

You don't do anything in the UIApplicationDelegate applicationDidEnterBackground: method.

Any task that is wrapped inside the "background task" calls will be allowed to keep running when the app enters the background.

Here's the really important part - the task only gets 10 minutes maximum. If it is still running after 10 minutes your app will be terminated. The expiration handler gives you a few seconds to cleanly end the task before the app is terminated uncleanly.