It's just like someone/somebody. They really should be two words and there is no technical reason they can't, but we write them as one for absolutely no reason. To be honest I write 'can not' everytime, but my computer changes it to 'cannot'. I really don't see the point in writing it as one word when the abbreviation "can't" already exists, but that's the convention and if that's what your teacher says you should stick with it.

In American English they're both acceptable. http://grammar.quickanddirtytips.com/cannot-vs-can-not.aspx This article explains when and why you might choose one over the other: http://www.write.com/writing-resources-articles/general-writing/word-choice/cannot-versus-can-not/

Every time I use cannot in speaking , I am trying to stress my denial and negative attitude affirmatively . aside from that, I cannot tell its noticeable distinction .

Or send your teacher here and Jmat and I will set him straight ;-)

Short answer: it does exist and it is not ungrammatical, however it is not commonly used, and you should just use 'cannot' and not waste your breath arguing with your teacher ;-)