The radial growth of univalent functions

Let f(z) belong to the well-known class S of functions univalent in the unit disk. It is shown that, in a classical result of Spencer (Trans. Amer. Math. Soc. 48 (1940) 418), this lim-inf condition cannot be replaced by a lim-sup condition. There is a function f in S for which the set for which the lim-sup is positive is uncountably dense in every interval and its complement is of Baire Category I. Such a function cannot be close-to-convex.