Response to referee after rejection

This is a waste of your time, and, more importantly, a waste of the time of people who already selflessly gave away their time working to give you feedback that helps you improve your paper, and to help the peer-review system function.

The best thing you can do is not ask the editor and referees to spend any more time on your paper, unless you are contesting the rejection decision in good faith. And use their feedback to improve your paper so that other referees and readers don’t reach the same erroneous conclusion that this referee did.

I'll disagree with Dan Romik about this: I'd think it's totally fair to send the referee an rebuttal through the editor in charge of the manuscript. (And I say that as the Editor-in-Chief of a journal.) I also agree with the OP's comment on the other answer: "when I review a paper and raise questions or objections, I often lament that my questions are not resolved or explained when the paper is rejected. As a referee, I am constantly learning/thinking about new problems from new perspective: I think there is some (potential) value for a referee in having their questions address (and thus, helping their understanding of a new development they express genuine interested in)."

I do think, however, that the better approach is to actually think about why the reviewer might have gotten the wrong idea about the underlying assumption. What was it that didn't explain the issue clearly and allowed the reviewer to go in the wrong direction with their review? Presumably, if a reviewer who is experienced in this field didn't get this important point, then others might as well. So the productive approach is to critically revisit your writing and , in any future manuscript on the topic, add sufficient discussion to ensure that this critical point is actually addressed and unambiguous. This way, you're not just correcting the opinion of that single reviewer, but of potentially any number of other readers who might have also wondered why you are writing a paper on an unfounded assumption.