Definable function ($\mathrm{graph}(f)\text{ is definable}$)

A function $f:M^n\to M$ is a definable function if its graph of a function is a definable set. Equivalently, a first-order formula $\theta (\mathbf{x},y)$ defines exactly the pairs with $y=f(\mathbf{x})$. Totality and uniqueness mean $M\models \forall \mathbf{x}\exists !y\theta (\mathbf{x},y)$. The same definition with restricted domains gives definable partial functions. A function need not be a symbol of the language to be definable; the graph can be specified through the existing relations and operations.