Abstract

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 $x \not\in\lbrack -\pi,\pi\rbrack$