All personnel, except those specifically designated by the provisions of this chapter as removable at the discretion of the mayor, holding administrative office at the time the mayor-council form of government becomes effective shall continue in office and in the performance of their duties until other provisions have been made by law for the performance of such duties or the discontinuance of such office. If any office, department, or agency of the city shall be abolished by this chapter, the powers conferred and the duties imposed upon such units by the laws of the state shall thereafter be exercised and discharged by the office, department, or agency designated by the council.