Why won't my Nikon D3100 autofocus in video mode?

I bought my camera yesterday and it is supposed to continuously autofocus in video mode, but no matter what settings I choose it just won't do it. The only way I can get it to focus is by constantly half-pressing the shutter button.

Added (1). Yes, I have read the manual and searched the net.

Added (2). According to the manual it does HD recording with continuous autofocus. I saw it in action in the shop.

Have you read the owner's manual? Details will be there.

2 things they won't tell you in the handbook or on the sales pamphlet:

1: AF and video don't go.AF is fine for static subjects, or getting one frame in a burst of moving subjects in focus. It does not work for contiguous frames as in video.

2: MF with a big sensor and small low res LCD don't go.

The video from DSLRs can be great, but you have to be on your game. It's not quite sorted enough for non-professional use.

Your D3100 doesn't autofocus during video mode because it is a dSLR, not a digital video camera.

"… It does HD recording with continuous auto focus. I saw it in action in the shop."

So take it back to the shop and ask them to help you set up your camera to do what it did in the shop. They'll be more than happy to help you since their goal is a satisfied customer who comes back and makes additional purchases.