difference between carry on, continue, and go on

• Both expressions mean the exact same thing in that you would be asking someone to continue speaking, whether they were interrupted by someone else ( for example, someone poking their head in to give a message or ask a question during a business meeting) or the speaker stopped (for example out of concern that there was no longer time to finish). In either case you could use informal expressions such as "go on" or "carry on" or a more formal, polite term such as "please continue". In a formal setting such as a business meeting it would be much more appropriate to say "please, continue" rather than "go on" or "carry on" which you would use informally with friends and family.       

• Both mean to persist in an action.They could have 2 different meanings when referring to speech.
  
  "to carry on " or " to continue " could mean to resume speaking after an interruption.
  If 2 persons are in the course of a conversation then a 3rd distracting person interfered joining the conversation for a moment ;once that person finishes one of the 2 main persons asks the other to "carry on" or "to continue" the previously interrupted speech.
  
  They both could also mean to go on uninterruptedly keeping the course of conversation. 
  
  However there is a slight difference as follows:
  2 persons in a conversation and a phone call interrupts their course.
  The one takes the call then asks the other thereafter to carry on his speech. 
  "Sorry to interrupt you , carry on".
  It means start there where you were interrupted and resume. 
  
  If the speaking person stops for a while to do something not being interrupted by an external cause ,but to pursue an action himself, in that case the other person could tell him :
  " go on " or" continue". 
  That means continue what you have been doing uninterruptedly by an external element but shortly ceased by you.
  
  Notice that the time span between the interruption and resuming the action is longer in the case of using " carry on " than when using "continue". When using the later there is a continuous progression 
  with a possible interruption that is shorter in time than that one demanding to "carry on" the speech.