Christmas and New Year period in Japan

I will spend my Christmas and New Year holidays in Japan this year. I've never been there in that period, but I remember I read on a tourist guide that on the very first days of January almost everything (department stores, shops, restaurants, etc.) are closed. Is it true? I am going to spend my holidays mainly in Tokyo and cannot believe that everything is closed in that city! What about Konbini? Do they close on specific days or are open 24h 7/7 as always? Many thanks in advance for your reply!

Many stores are closed on January 1, but it seems that with every year, more stores are open. Many of the new shopping districts, like Roppongi Hills, Odaiba, etc. are open on January 1. Convenience stores are open. More information:
http://www.japan-guide.com/e/e2276.html

Great and helpful as usual: thank you Uji!