Of course, one-to-one functions have a different y value for all x values.
y=mx+b would be one function that suits your criteria.
Correct me if I'm wrong, but this would not always produce an integer y for any real positive x.
Is there a proof or theorem somewhere that says or implies that if "the domain is an uncountable set and the range is countable" then there can be no "one to one map " ? I'd be very interested in that.
Thanks