We can enable a fail-safe operator feature that will receive all emails in the event that SQL Server Agent cannot communicate with the system tables in the msdb database. This is accomplished by recording the information about the fail-safe operator in the registry. The fail-safe operator will also receive notifications if you have scheduled operators to only receive notifications during certain time periods and a notification occurs outside of that range. For example, if no operators are set to receive notifications on Sunday, the notification will automatically go to the fail-safe operator. 

To enable a fail-safe operator, right-click SQL Server Agent and select Properties from the context menu. Then select the Alert System page as shown in the following image. 

Select the Enable fail-safe operator checkbox and then choose the appropriate operator and notification method. Click OK to save the changes.