Definability of predicates in Continuous first order logic is (usually) defined as follow:
A predicate $P:M^n \to [0,1]$ is called definable over $\mathcal{M}$ iff there exists a sequence $(\phi_k(x)|k\geq1)$ of $L(M)$- formulae s,t, the formulae $\phi_k(x)$ converge to $P(x)$ uniformly on $M^n$; i.e.,
$\forall \epsilon > 0 \quad \exists N \quad \forall k \geq N \quad \forall x\in M^n \quad (|\phi_k^{\mathcal{M}}(x) - P(x)|\leq \epsilon)$
I understand the definition, but can't think of any undefinable predicates under this definition. Is there any classic, or maybe modern that I do not know about, examples for that?
If you take a discrete structure $M$ then the indicator function of a subset of $M$ is a definable predicate if and only if that subset is definable, so you could take $M$ to be $\omega$ with just the discrete metric (i.e., equality) and then the function $$P(x) = \begin{cases} 1 & x\text{ even} \\ 0 & x\text{ odd} \end{cases}$$ will not be a definable predicate.