pymongo : to check if we have connected to MongoDB database

You will probably have to use the authenticate method anyway inside connection function, so you could do it as following:

In [12]: client = MongoClient(DB_MACHINE, DB_PORT)
In [13]: db = client.db_name
In [16]: if db.authenticate(DB_USER, DB_PASS, source=DB_SOURCE):
             # authenticated, do ...
         else:
             # not authenticated, not connected, do something else