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Actually, f(0) is well-defined. If q is positive, then gcd(0,q) = q, so f(0) = 1.

And when q is negative? Don't we have 1 = 3/3 = f(0/3) = f(0/-3) = 3/-3 = -1?

This same problem will occur everywhere negative, though. I wasn't thinking about it; I was just being sloppy.