I suppose the task has to say something about how fast the air around the finger gets replenished. If we assume that the air directly outside of the finger is always at 800 degrees C, so that it is instantly replenished as fast as it gets cooled down by the finger, the finger would also have a temperature at 800 degrees C furthest out. Only if you suppose that the air is not instantly replenished so that it can get cooled down, it will be possible to keep the finger in the flame for a while without getting burnt (well it will be otherwise too; we just can't show it analytically!).
By the way: