If f is bounded and
holomorphic in the unit ball in Cn then it has radial limits at almost all points of the
boundary of the ball. More is true; for example, f will have limits almost everywhere
with respect to arclength on any arc that forms part of the boundary of an
anlaytic disc. Motivated by these considerations we consider an n-dimensional
torus in the boundary of the ball and ask if there are growth conditions less
restrictive than boundedness that imply the existence of radial limits on this
torus. We show that the answer is no for some of the standard function
classes. For example, we show that there is holomorphic function of bounded
mean oscillation in the ball that has a finite radial limit at no point of the
torus.
