My iPhone 5 says it's not connected to the internet, but it is?

It won't let me check weather, go on Safari, and Siri won't respond at all. Everytime I load one of those, it always says the same message. I got it almost a month ago, and I would hate to take it to the Apple/Verizon store to get it checked out. I already turned it off and back on for a few minutes, still didn't work. But the thing is, it IS connected to the internet! Help, please?

Added (1). @zuma, I tried that, still doesn't work.

Added (2). @zuma, I tried that, still doesn't work.

Go to Settings, General, Reset, Reset Network Settings. Wait a few minutes and shut your phone completely off. Then wait 2 minutes & turn it on. See if that helps. It should.

Call your carrier, it porbably has nothing to do with the phone.