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This question is asking about school reports, letters to members of Congress, apology letters, etc.

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    Stick with cannot. Can't tends to be informal, and can not is usually considered a typo. – ralph.m Dec 10 '18 at 22:24
  • @ralph.m Or it can be a rhetorical way of underlining the impossibility. – Tuffy Dec 10 '18 at 22:29
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You should avoid using contractions like "can't", so only use the long versions of negative construction. "Cannot" can be written in almost all cases but it's also possible to use "can not". You generally need to use it only if it's a part of some other construction, such as "not only...but also."

Example: "The company’s new product can not only reduce emissions, but also trap some of the existing greenhouse gasses."

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