You can try use Log Analytics workspace and configure an alert email when the function app is published.
First, create a Log Analytics workspace, then in azure portal -> nav to the Log Analytics workspace
-> Logs, and write the following query(Note: the caller is your azure account(email address)):
AzureActivity
| where Caller == "xxx@xxx.com"
| where OperationName == "Create Deployment"
| where OperationNameValue == "Microsoft.Resources/deployments/write"
Then click "New alert rule" to create the rule.
The screenshot as below:

Then in the "create rule" page, click Condition to configure details. The screenshot as below and please configure the values as per your need:

After you configured the rule, when you publish a new function app, you will receive an alert email(if you configured the alert sent to an email).