Continuity

Abstract

Continuity and constructivity have an intimate relationship, which has been recognized for nearly a century, studied for half a century, and is still fruitful today. The connection is easy to see: If we are to be able to compute some number F(x) for a real number x, we must be able to proceed with the computation given approximations to x. Our method for computing F(x) should then yield approximations to F(x). Evidently F will then be continuous. Recognition of this idea goes back at least to Hadamard, who formulated three conditions for a problem in differential equations to be “well-posed” (see e.g. Courant-Hilbert [1953], p. 227):

• (i) existence of the solution

• (ii) uniqueness of the solution

• (iii) continuous dependence of the solution on parameters (initial conditions or boundary conditions, for example).