if you bought your iPhone 5 from the Apple store and it is still in warranty, I would return it.
you could also log in to Apple's website and live chat support.
in my experience of iPhones I've always liked the iPhone 5. It's a robust little phone that fits perfect in one hand.
Here is one thing I would try.
Go to Settings > General > Reset > Reset Network Settings
Hold the home button and lock button down until the iPhone restarts and keep on holding.
you will see the Apple logo appear once and then disappear.
when it appears for the 2nd time, let go of the buttons and let it boot normally.