Keep on: "to continue doing something, or to do something many times."
Go on: "a) to continue doing something or being in a situation. b) to continue without stopping".
From Longman Dictionary of Contemporary English
Learn English – difference between “keep on doing” and “go on doing”
differencesmeaning
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Best Answer
Per my comment to the question, in most contexts, all three of keep on / go on / continue [present participle] are synonymous and interchangeable (arguably I've given them in descending order of "informality").
But in this context I would very much prefer keep on (and definitely wouldn't use continue)...
Maybe it's just a personal thing, but I feel to keep on has more overtones of be persistent, dogged, whereas being told to go on often means little more than "don't stop".
There are probably other contexts where one term is commonly understood to be more (or less) suitable than the others, but nothing else comes to mind at the moment. So comments welcome.