Hi.. Thanks for the query.. As per your question you should have not tried to pull the tooth by yourself and whatever you have pulled out must be just the top part of tooth called as tooth crown(that portion of tooth which is seen in the mouth) and the roots which are present covered with the gums and underlying jaw bone must be definitely present inside.. The remaining tooth portion can get infected as food debris will get deposited in it and will serve as source of infection and once it get infected there will be pain.. You should not at all keep it like this.. I would suggest you to go to a dentist and get evaluated. He will do a clinical check-up as well as take an IOPAR (x ray) to check the status of remaining tooth structure as a root canal treated tooth becomes already weakened and may crush or break while extraction.. It will be advisable for you to get the tooth removed and get it replaced with artificial teeth(fixed or removable will be decided by the dentist after thorough check-up of the surrounding structures like gums, underlying bone, and adjacent teeth). To avoid infection you should always brush your teeth two times a day (morning and night) and gargle using a mouthwash after meals. But removal of your broken tooth is must for preventing future problems of infection and pain. Hope your query is solved.. Thanks and regards..