Implied easement is just like its names suggests. It is implied without any legal documents. Prior use by the city may be enough to acquire an implied easement. If the city performed any public service on the road, like you suggested, it may be enough to acquire an easement. In your case, it is not clear how far back the city took control of the road. If I were the city attorney I would have at least three argument against your claim. One is the implied easement. Second is the easement by necessity. Third may be an easement by estoppel. Obviously you had notice of the public work and all these arguments may work against you. Another possibility is that an easement may have been reserved for public work before the land was divided years ago, as all cities do. If you are absolutely positive there was no reservation or grant of an easement, then the city's action is eminent domain and you are entitled to compensation, although you may not get rid of the easement now. As an easement holder, city has a duty to maintain and not to cause damage to the land above. You may have a claim to have the erosion damage repaired by the city. You should consult a local eminent domain attorney for help.