I have the same problem in UK and my iPhone 5 was working fine with 3G and I'm not a 4G subscriber but suddenly a week ago my phone started to switch between 3G and Edge. Even restarting the phone, Airplane mode or hard reset did not get my 3G back. The only way I can get the 3g is by removing my sim card and installing another one then installing back my original sim card and after a while of 3G the Edge (E) returns like a ghost.
I had a sim replacement but that did not fix the problem. I used my old iphone 4 and the problem was gone!
return to iPhone 5 and the Edge again !
Then I found the problem !
It's the carrier, not the phone!
Every network has two coverage 3G and Edge, in iphone 4 and 4s, the option of having 3G or Edge is set so iphone 4s or 4 seem to track only 3G network when 3G toggle is ON. In iphone 5, the phone work both the 3G and Edge,which ever is available but set the priority for 3G in the first run of the simcard. Now what is happening with me is that the 3G network is having a continuos blackout moments and therefore the phone switch to the Edge network Automatically and when you go to an area with only 3G coverage the phone will be forced back to 3G but loses it again when the 3G is stopped for some reasons.
Also it seems that this problem is happening with carriers with 4G or LTE network.
So I'm trying this solution and hope it will work, I turned off the LTE toggle and turned off the automatic selection of the carrier and choosed the 3G network and i will see tomorrow if that would work !