Every person elected or appointed to any office under this Charter shall, before they enter upon the duties of such office, take and subscribe an oath of office, and file the same, certified by the officer administering the same, shall, before entering upon the duties of their respective offices, be required to execute such bonds to the City of Austin as the city council thereof may direct and approve for the faithful performance of the duties of such office, and such bonds may be increased or diminished at the pleasure of the council. If any officer fails to give the required bond, the city council shall declare the office vacant and proceed to fill the vacancy as herein provided.