P.S. Laplace wrote in 1798:
  "A luminous star, of the same density of Earth, and whose diameter
   should be two hundred and fifty times larger than that of the Sun
   would not in consequence of its attraction, allow any of its rays
   to arrive at us; it is therefore possible that the largest luminous
   bodies in the universe may, through this cause, be invisible."
 
_Gravitation_ by Misner, Thorne & Wheeler presents a dialog explaining
why black holes deserve their name.  (It is on pp 872--875 in the 1978
paperback edition, ISBN 0-7167-0344-0.)

As explained in D.03, light rays follow geodesics in spacetime.  To
describe things fully you need Eddington-Finkelstein coordinates.  In
these coordinates it's pretty easy to see there is a 'surface of last
influence'.  In fact, page 873 of MTW has a pretty good graphic showing
just that.  The surface of last influence is the 'birthpoint' of the
black hole.  It's also clear that in the normal sense of things, 'up'
doesn't exist on the surface of a black hole.  As a matter of fact,
black holes don't really have solid surfaces as you might be thinking.
 
Black holes have horizons, but that's a region in space, not a solid
surface.  If you draw various world lines of observers travelling in and
around black holes you will see that the light cones of observers who
don't cross the event horizon have some segment of those cones above the
horizon.  Those observers who do cross the event horizon of a black hole
are constrained to fall toward the center eventually.  There simply are
not any geodesics that cross the horizon in the outward direction.
 
At the center there is a region of infinite density and zero volume
where everything ends up.  This is a problem in the classical
understanding of black holes.

Recent attempts to understand black holes on a quantum level have
indicated that they radiate thermally (they have a finite temperature,
though one incredibly low if the black hole is of reasonable size) that
is proportional to the gradient of the gravity field.  This is due to
the capture of virtual particles decaying from the vacuum at the
horizon.  These are created in pairs and one of them is caught in the
black hole and the other is radiated externally.  This has been
interpreted by Hawking as a tunneling effect and as a form of Unruh
radiation.  This may give some clever and knowledgeable researcher
enough information to figure out what's happening at the center someday.
 
Another way to think about things is to consider basic geometry.  The
surface area of a ball is related to its diameter by pi.  A = pi*d^2.
But any gravitating body distorts space so that a light beam travelling
through the center of the body measures a diameter slightly larger than
that indicated by the surface from which it is measured.  In the case of
a black hole the diameter measured in this way is infinite, while the
surface area is finite.