[math-fun] Does such an curve exist in 3-space?

Let f: R —> R^3 be a C^oo one-to-one mapping of the reals into 3-space. (For convenience, assume WLOG that ||f'(t)|| is never 0.) Suppose further that a) The image f(R) is closed and bounded in R^3; and b) If for some sequence t_j in R we have lim f(t_j) = f(t) j—>oo for some t in R, then we also have convergence of the tangent vectors: lim f'(t_j) = f'(t) j—>oo ------------------------------------------------- Question: Does there exist such a curve as f ??? (Note: If not for b), an easy example would be for the image f(R) to be a figure-8.) —Dan