“I'm sorry, you're breaking up.”
You're talking on your mobile phone to your boss. He is saying something, but you can't hear what it is because the reception on your phone is bad. You didn't hear some of what he said, so you say this.
I'm sorry, you're breaking up.
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This is a phrase you use when part of a conversation over mobile phone gets cut off because of bad reception:
Hey, you're breaking up. Let me call you back in a few minutes.
You can also say that "the connection is breaking up".
When you're speaking with someone you don't know well, it can be polite to say "I'm sorry" before asking a question. For example:
I'm sorry. What was that?
I'm sorry. Do you know where the restrooms are?
You can also apologize for problems that aren't your fault.
I'm sorry. I didn't catch what you said.
The implied question here is "Can you say that again?"
The reason for apologizing in these situations is that you're supposed to listen and pay attention carefully when speaking with people who you don't know well. However, when someone says "I'm sorry" in this way it probably doesn't mean that they feel truly sorry.