No. A 1/2 gallon is too small for anything at all, and if I had my way these "tanks" would not be in production for fish.
A Betta is a delicate tropical fish, in an unfiltered, unheated tank like that it would be prone to all sorts of things including fin rot and parasites as well as becoming quite lethargic and bored in such a tiny cold space.
For a happy a Betta you need something at least like this with a heater added (25-50w):
http://www.petco.com/product/13809/Marineland-Eclipse-Corner-5-Aquarium-Kit.aspx
And it would have to be fully cycled before hand
http://www.theaquariumwiki.com/The_Nitrogen_Cycle
Edit: First L - well done for believing in stupid myths concocted by pet stores. Bettas do NOT come from puddles, they originate from vast rice paddies acres wide, as well as forest streams and marsh bogs. NOT PUDDLES!!! And those in Thailand would find it difficult to believe they live in a sub-tropical/temperate area too, these are tropical fish needing at least 76-80F to be healthy and active.
If they can only "manage" a gallon how come mine lived in 8 gallons to themselves quite happily for over 3 years? Load of bull.