Whenever all or any part of any township, improvement district, or other governmental unit is annexed to any city, such township, improvement district or other governmental unit may, continue to furnish services for the year for which taxes have been levied or collected in those areas of the district annexed, or in the alternative, shall surrender the taxes collected to the annexing city to be used specifically for the purposes for which the tax was collected. No improvement district shall continue to make a levy for its general fund upon territory annexed to any city, from and after the effective date of such annexation.