<< More of an intellectual curiosity question than anything useful. Is there a "nice" function that is one-to-one and continuous that maps the closed interval [0, 1] to the open interval (0, 1)? Ideally, such a "nice" function would be not a piecewise function or a power series and something that could be implemented with standard library functions. If such a thing doesn't exist, is there at least a "not nice" function that is one-to-one and maps the intervals? If not, why not?
Any continuous function f: [0,1] -> (0,1) cannot be *onto*, just because the continuous image of a compact set is compact. Since a continuous map also preserves connectedness, f([0,1]) will be a closed interval [a,b] with 0 < a <= b < 1 (or a point if a = b). -------------------------------------------------------------------------- There is an interesting curiosity, however, that there *does* exist such a function, in fact a *homeomorphism*, from the rational numbers in [0,1] to the rational numbers in (0,1): h: Q \int [0,1] -> Q \int (0,1) where these spaces are topologized as subspaces of R. I'll leave it as a puzzle how to define such an h (for those who haven't already seen it). --Dan