Oh yeah! I forgot they told me that too! I about fell over and died laughing when they told me that.
That's a lie, because not only can I still see the wifi antenna icon when I click the phone ON to the lockscreen, when I'm using Pandora or any other streaming music over wifi it would stop working every time the screen shut off. But I know, they'll tell us "it wont' shut off when something is using Wifi", yeah, ok and I have some land in Florida I want to sell you too.....
Wifi takes anywhere from 2-10 seconds to connect depending on how strong the signal is. It takes about 1/2 a second to turn the phone on. It's impossible for wifi to connect that fast. If wifi is not connected, there is no wifi icon.
Wifi doesn't shut off when the phone goes into lock mode. If wifi disconnected when in Lockscreen mode you would NOT see the wifi icon when on the lockscreen as indicated below.
Here are three videos, (on my YT) one of me turning the phone on and you'll notice the wifi indicator is connected already-- it does NOT connect when we turn the phone on, or take it out of screenlock
Turning phone on to lockscreen
Here is a video of me showing how long it takes wifi to actually turn on, and I have a strong signal, so it connects fast. If I were connecting to public wifi it would take up to 10 seconds
So if wifi is off in lockscreen mode, how on earth do I get FB notifications? I have 3G turned off and the phone is locked, but whatdyaknow! A FB notification!
I also texted myself from my son's phone -- who has NO service, it's strictly wifi. So any text messages are sent via iMessage--- OVER WIFI. If wifi is turned off, HOW did I get the messages? Is AT&T going to say it goes over 2G? (which is impossible because iMessage only works with Wifi [until iOS6]) Well then if all that data got sent over 2G, I'd STILL have exhorbitant data usage every month because I NEVER turn my phone off! EVER!
I WILL figure this out, and if it really was accumulated data over the period of the 4 months then so be it, if I cannot find any plausible explanation, I will dispute it.