You should contact local personal injury attorney ASAP because suing the City of San Francisco can be tricky in terms of providing the requited notice in a timely manner. You miss the deadline and you cannot sue them.
DISCLAIMER: David J. McCormick is licensed to practice law in the State of Wisconsin and this answer is being provided for informational purposes only because the laws of your jurisdiction may differ. This answer based on general legal principles and is not intended for the purpose of providing specific legal advice or opinions. Under no circumstances does this answer constitute the establishment of an attorney-client relationship.
While I cannot give you legal advice in this forum, you may have a case. Contract a local personal injury attorney for a consultation. You should not wait. Suing a governmental agency is a little more complex than suing an individual. There are additional steps required.
THIS RESPONSE IS INTENDED TO CONVEY GENERAL INFORMATION ONLY. IT SHOULD NOT BE RELIED UPON OR TAKEN AS LEGAL ADVICE. FURTHER, THIS RESPONSE IS NOT INTENDED TO AND DOES NOT CREATE AN ATTORNEY-CLIENT RELATIONSHIP.
Public entities in California can be held liable for a "dangerous condition of public property", however, there is a short tort claims statute of six (6) months from the date of the incident. That requires that you file a claim with the City of San Francisco within that time or your right to sue will be lost. Also, there are various "immunities" that apply to public entities that have to be overcome in order to prevail. These are difficult matters and you should talk to an experienced attorney in San Francisco. I recommend Chris Dolan of the Dolan Law Firm in San Francisco.
Whether the city is liable for your injuries depends on a number of factors. As other attorneys have mentioned, you have six months from the date of your injuries to file a claim against the city. You should talk with an attorney who is experienced in handling personal injury matters. There are many fine attorneys on AVVO.
You may have a case against the city, but you should contact an attorney, as you only have 6 months to fail a claim against a government entity. Below is some more information about dangerous road defects and accidents.