 These functions clearly cannot be extended to arbitrary real numbers without making them partial, as doing so would violate the Extreme value theorem.Since that sort of behaviour could be considered pathological, it is natural to insist that a function should only be considered total if it is total over "all" real numbers, not just the computable ones.In the event that one is unhappy with using Turing Machines (on the grounds that they are low level and somewhat arbitrary), there is a "realisability topos" called the Kleene-Vesley topos in which one can reduce "computable analysis" to "constructive analysis".