Abstract: A discrete function theory is found based on the continuous function theory and the discretization of real numbers, which is used to express the classical general relativity and analyse the collapse of a uniform-density ball of dust. It i s shown that the intrinsic and extrinsic geometrices do not actually match at th e join between the Friedmann interior and the Schwarzschild exterior for the Opp enheimer and Snyder solution as a continuum. With the discrete function theory, we can extend and discretize the solution to form a complete non-collapsing solution of the ball of dust, thereby eliminate the gravitational singularity and reveal the deep nature of the discretization of spacetime.

Keywords: 
• discrete real number /
• discrete spacetime /
• general relativity /
• Oppenheimer and Sn yder solution /
• singularity-free