You can keep going as you said and improvise.Many actors improvise and continue with the action of the play.Like if you forget your lines,I think you should stay with your objective and improvise and usually your fellow actors will try to help you.You shouldn't break character,because it may take the audience out of the circumstances of the play and makes the play appear amateurish.If your sword breaks in half and you have to kill Mercrutio who has a sword that works,you will have to find a believable action to achieve your objective,like picking up a chair and using that instead.Don't really hit him with the chair,but use your stage combat and act it out.Be careful.Try to make the action truthful and work with the script.If some technical flaws occur,I think you should stick with your objective and improvise again. If the mike stops working,use your voice production and articulate.Microphones were not used in the early years of theatre.Unless it is something completely out of your hands,then the director or stage manager probably can help get the play back on track.