Continuous If-Then Statements Are Computable

Abstract

In many practical situations, we must compute the value of an if-then expression f(x) defined as “if c(x) ≥ 0 then f ₊(x) else f ₋(x)”, where f ₊(x), f ₋(x), and c(x) are computable functions. The value f(x) cannot be computed directly, since in general, it is not possible to check whether a given real number c(x) is non-negative or non-positive. Similarly, it is not possible to compute the value f(x) if the if-then function is discontinuous, i.e., when f ₊(x ₀) ≠ f ₋(x ₀) for some x ₀ for which c(x ₀) = 0.

In this paper, we show that if the if-then expression is continuous, then we can effectively compute f(x).