In my Azure Function I am using a Library which establishes a connection to an SQL server via the ConnectionString from the ConfigurationManager like this:
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I had this same issue and am using .net standard (as opposed to core). I added my settings to the Application Settings section of my Azure function (in Azure Portal):-
I then downloaded a zip of the function:-
Included in this download is a copy of local.settings.json that includes my app settings in the correct json format. I then access them via ConfigurationManager.Appsettings["mysetting"]