It's a Kohlrabi....and yes, they can have the whole thing.
You can actually eat the bulb part...peel like a potato, then cut into fine chip like strips, boil in water for around 10-15 min. until tender. Add a white sauce with a bit of lemon and parsley.....yum!
The good bit is - kohlrabi last quite a while in the fridge, so you don't have to feed the bulb straight away.