Let f be a homeomorphism of the unit sphere Sⁿ and (f) be the Douady-Earle extension of f to the closed unit ball B^{n+1} bounded by Sⁿ. We show that if f is C¹ in a neighborhood of a point p Sⁿ and D f (p) is non-singular, then (f) is C¹ in a neighborhood of p in B^{n+1}. Moreover, when n=1, if f is orientation-preserving, then the complex derivative of (f) at p is equal to f^ (p), and z (f) (z) and z (f) (z) converge to f^ (p) and 0, respectively, as z p; if f is orientation-reversing, then the anticomplex derivative of (f) at p is equal to f^ (p), and z (f) (z) and z (f) (z) converge to 0 and f^ (p), respectively, as z p.
Hu et al. (Thu,) studied this question.