what's the difference between "check up' and "check on"?
They seem have the same meaning. Both are check something carefully. Are they really the same? If not, what's the difference? How to use them seperately.

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Answer

Hmm - I have to think about this.
If you say -"he's been checking up on me", this means he has been keeping a regular watch on me.  It implies suspicion.
"I'll have to check on that" really means the same as "check" by itself.  In fact, the extra "on" doesn't sound good.  
I can't think of any use of the phrasal verb "check up". I think you can safely ignore it.
If we go to the doctor for a routine appointment, we call it "a check-up" - this is a noun.   
Hope that helps, Michael