Have you had a chance to review the Monitoring Cloud Functions documentation page (
https://cloud.google.com/functions/docs/monitoring/)? The kind of notifications you're looking for can be accomplished via Error Reporting, or using one of the
built-in Cloud Functions metrics, or logging your error and using Logs-based Metrics. There are also a bunch of (non-Google) tutorials online on how to monitor Cloud Functions with Stackdriver. Which of these is "best" depends heavily on your use case — how you plan to track your errors, whether you care about individual errors or a spike in error rate, how fast do you want to respond to an error, etc.