Continuous If-Then Statements Are Computable
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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).
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