Bug in live chess

It told that he/she will auto resign in 0:01 and then it became 0:00 but the clock was still going on and he/she still did not auto resign 
it is a bug, right?

oh and just noticed when he had 0:10.5 sec it auto resigned 

It's not really a bug. The auto-resign countdown kicks in when:

a) The opponent is not making their first move.

b) The opponent's network has disconnected.

If they correct the issue before the allotted time has lapsed, the timer stops and the game continues.

If the game is already underway, such as yours was, and the opponent has not made a move for a certain long period of time, it's possible they may be auto-resigned without the timer. It's unclear exactly what the site's protocols are on this matter - people are always complaining on the forums about unfair cutoffs. I've never had much of a problem with it.

No well you are trying to say a diff thing.

I am saying that he did not play for a long time and the auto-resign timer started and after a min or so it ended but the real clock, the chess real online clock was still going and the game was too, but it was supposed to end it but after 1 min the game ended  (i.e. when the timer was on 0.10.5 or so) 

Perhaps it's too give the player a little more time to make their move after they've been warned.

I wouldn't pay it much mind, the auto-resign timer has always had a mind of its own.