Well, first off, and not to be mean, but a half gallon tank really is not big enough for a betta to live in. I would go for at least a 3 gallon tank. Some may argue that even that isn't big enough, but I think its the bare minimum a fish like a betta should be housed in. They also need a source of heat or they will start to shut down aka not take food, stay in one place in the tank, that sort of thing. There should be a tank heater in there because they are tropical fish after all.
Now that we have that out of the way, a healthy betta should be swimming around in their cup/bowl or what ever they are in at the store and engage in flaring. Their fins shouldn't be clamped and they should have bright and alert eyes.