For realanalytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function
given by ƒ(x) = x3. This function is clearly injective, but its derivative is 0 at x = 0, and its inverse is not analytic, or even differentiable, on the whole interval (−1, 1). Consequently, if we enlarge the domain to an open subset G of the complex plane, it must fail to be injective; and this is the case, since (for example) f(εω) = f(ε) (where ω is a primitive cube root of unity and ε is a positive real number smaller than the radius of G as a neighbourhood of 0).
^John B. Conway (1996) Functions of One Complex Variable II, chapter 14: Conformal equivalence for simply connected regions, page 32, Springer-Verlag, New York, ISBN0-387-94460-5. Definition 1.12: "A function on an open set is univalent if it is analytic and one-to-one."