Can an abscess continue even after a root canal treatment?

Yes and No: If the root canal was successful the abscess should heal by its own accord as the cause of the infection has been eradicated&this can take up to a year for complete healing of the apical area. If it persist then the tooth may be fractured. If this is the case it may have been fractured before the rct. See your dentist/see a specialist for a second opinion.

Yes: Despite your dentist's best efforts to clean and seal a tooth, new infections might emerge after a root canal caused by more than the normally anticipated number of root canals in a tooth (leaving one of them uncleaned), a breakdown of the root canal sealing material a defective or inadequate restoration that has allowed bacteria to get past the restoration and reinfect the tooth.