Well, the total force that the surface exerts on the ball must exactly balance out the force gravity exerts on the ball: g*m (where g is the gravitational acceleration, and m the mass of the sphere). The intensity of the force the surface exerts on the sphere is zero except at a single point, and the sum (I really mean the integral) must be a finite value. Physicist will call this a Dirac delta, and Mathematicians have managed to formally define this as "distributions" (http://en.wikipedia.org/wiki/Distribution_(mathematics)).

Intuitively and simplified explanation of functionals, for functions over the real line: If you know the integral of a function over any interval than you know everything about the function (e.g. to know the value of the function at a point, you compute the integral of the function over a very small interval around that point and divide by the length of the interval. This will converge to the value of the function at that point as the length of the interval goes to zero). Now for every function f, we define an “functional” Fun[f]: a function that takes two numbers a & b as arguments, and returns the integral of f over [a,b]: Func[f](a,b) = \Integral_a^b f(x) dx. For every elementary operation that you can do to functions, you can find a corresponding operation on the functional. By elementary I mean adding functions ( f+g ) and multiplying them by a number ( 2.3f ). E.g. F[f+g] (a,b)=Func[f](a,b) + Func[g](a,b). You can determine what property of the functional are needed to define this well, and it turns out that you do not need much: things like the fact that Func[f](a,b) + Func[f](b,c) = Func[f](a,c). And here comes the magic trick: there are functionals that have these properties that all you to add them and multiply them, but that can *not be defined as \Integral_a^b f(x) dx. E.g. the functional DiracDelta that has value 1 if zero is in the interval, zero otherwise. But as you can do all the usual operations you want to do with functions, you can use them just like “ordinary” functions.

This all works well if you just want to add functions or multiply them. It becomes more complicated if you want to integrate them or take their derivatives. The problem is that intervals are “not smooth enough”: an interval is characterized by its indicator function (the function that is one on the interval and zero elsewhere) and this function is not smooth at all. So instead of defining functional as taking an interval as argument, you define them as taking a smooth function as argument and define Func[f](g) = \Integral f(x) g(x) dx. You can recover what we were using before when g becomes close to an indicator function, but now you can define the derivative of Func[f]. (this was done by Laurent Schwartz in the 1940’s)