(a) Where an appeal lies on behalf of the defendant or the people, it may be taken by the defendant or his or her counsel, or by counsel for the people, in the manner provided in rules adopted by the Judicial Council.
(b) When upon any plea a judgment of death is rendered, an appeal is automatically taken by the defendant without any action by him or her or his or her counsel. The defendant’s trial counsel, whether retained by the defendant or court appointed, shall continue to represent the defendant until completing the additional duties set forth in paragraph (1) of subdivision (e) of Section 1240.1.