We prove that no restriction of the sine function to any (open and nonempty) interval is definable in $\langle\mathbf{R}, +, \cdot, , and that no restriction of the exponential function to an (open and nonempty) interval is definable in $\langle \mathbf{R}, +, \cdot, , where $\sin_0(x) = \sin(x)$ for x ∈ [ -π,π], and $\sin_0(x) = 0$ for all $x \not\in\lbrack -\pi,\pi\rbrack$.

We investigate Turing's contributions to computability theory for real numbers and real functions presented in [22, 24, 26]. In particular, it is shown how two fundamental approaches to computable analysis, the so-called ‘Type-2 Theory of Effectivity' (TTE) and the ‘realRAM machine' model, have their foundations in Turing's work, in spite of the two incompatible notions of computability they involve. It is also shown, by contrast, how the modern conceptual tools provided by these two paradigms allow a systematic interpretation of Turing's (...) pioneering work in the subject. (shrink)