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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.

Here parameter-free definability is intended: parameters are not silently added to the first-order language. A subset $D\subseteq A^n$ is a definable set when some $L$-first-order formula $\phi (x_1,\dots ,x_n)$ satisfiesA definable function is one whose graph is a definable set. Thus $f:A^n\to A$ is definable when an $L$-formula $\theta (\mathbf{x},y)$ satisfiesIn particular $\mathcal{A}\models \forall \mathbf{x}\exists !y\theta (\mathbf{x},y)$. Definability over a parameter set $B$ permits parameters from $B$ in these first-order formulas, equivalently new constants naming those parameters. That is a different requirement: allowing a parameter naming an element would immediately define its singleton.