(a) Whenever a petition in writing shall be presented to the City Council, signed by the owner or owners of two-thirds of the frontage on any street, lane, or alley, or any portion thereof in the City asking that such street, lane, alley, or portion thereof be curbed or guttered, or that a sidewalk or the sidewalks thereof be constructed or repaired or paved, the City Council may order or direct that such improvements be made.

(b) The City Council without petition may make any or all of the improvements described in the preceding section and shall assess against the owner or owners of the lands or buildings abutting upon such street, lane, or alley and adjoining the part where such improvement may be made, whether made upon petition or otherwise, so much of the cost and expense of making such improvements as it shall judge such land and buildings to be benefited thereby; but in no case shall the cost and expense of making such improvements be assessed against the abutting owner on either side of such street or portion thereof in excess of one-half of such cost and expense in making any of the other improvements specified in the preceding section, and the City shall pay the remaining cost and expense of all improvements therein mentioned; provided that all assessments heretofore made shall remain in full force and effect until paid or abated.