Definable functions in FO(Z,+,≤), FO(R,+,≤) and FO(R,Z,+,≤) are exactly the piecewise-linear or piecewise-simple functions, and mixed sets coincide with semi-polinear sets.
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Extending the Ginsburg-Spanier Theorem to Functions and Mixed Arithmetic
Definable functions in FO(Z,+,≤), FO(R,+,≤) and FO(R,Z,+,≤) are exactly the piecewise-linear or piecewise-simple functions, and mixed sets coincide with semi-polinear sets.