0 present participle of apologize
1 to tell someone that you are sorry for having done something that has caused problems or unhappiness for them:
Trains may be subject to delay on the northern line - we apologize for any inconvenience caused.
Imagine the humiliation of having to apologize.
Apologizing for the error can be helpful in achieving some sense of resolution for us.
The book deals with four categories of conversational routines, viz. routines for thanking, apologizing, offering/requesting, and discourse marking.
This also explains the current and explicit practice of apologizing for any omission.