If you mess up, say so: "Oh, that's completely wrong. Huh. Ok, let's fix it." Again, these interviews are more about seeing how you reason about a problem. People make mistakes all the time - wrong assumptions, incomplete solutions, etc. What matters is what you do once you realize there's a problem.
I don't have any trouble admitting I made a mistake when I notice it, it's recovering from it. At that point I think about how I messed up, what I should have done, and how bad the rest of the interview will go.
In every day life, this is not the thought process I go through if I make a mistake. But in an interview the stakes are higher, you'll be quickly discarded if you don't do well.
In practice, you may lose even if the interviewer is wrong. I have. At some big-name companies.
Just for example, an interviewer at Apple was adamant that frequency can never be negative. Whatever happens to Fourier transform that integrates frequency from negative infinity to positive infinity.
This may very well be, but there's not much you can do about it at that point but continue to try to fix the problem. If you can't be amicable about it with the interviewer, then either they don't want to work you or you don't want to work with them (probably both). Might as well make the best of it.
Yeah, I think I once lost out on a Google interview in part because the interviewer lacked the math to follow my reasoning (also, they were attempting to get me to a different solution by applying criteria that my existing solution already adhered to better than what they wanted).
