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+(x0) ̸ = f−(x0) for some x0 for which c(x0) = 0. In this paper, we show that if the if-then expression is continuous, then we can effectively compute f(x). Practical need for if-then statements. In many practical situations, we have different models for describing a phenomenon: – a model f+(x) corresponding to the case when a certain constraint c(x) ≥ 0 is satisfied, and – a model f−(x) corresponding to the case when this constraint is not satisfied