In Alain M. Robert's book, he affirms that continuity is implicitly defined by S-continuity and give the following example: Let $I$ be a standard interval and $\mathcal{F}$ be the space of numerical functions on $I$ (hence $\mathcal{F}$ is standard too). Then, for a $f \in \mathcal{F}$, we could define continuity at $x$ using the Standardization Axiom.
$f$ is continuous at $x$ $\Leftrightarrow$ $(f,x) \in \{(f,x) \in \mathcal{F} \times I: \text{$f$ is S-continuous at $x$} \}^S$
The problem with this definition is that the interval $I$ must be standard in order to the set $\mathcal{F}\times I$ be standard too, since a standard set is required by the Standardization Axiom. But continuity is a general concept that should be applied to functions defined at non-standard intervals too (e.g: $I = [-\epsilon, \epsilon]$ with $\epsilon > 0$ infinitesimal and $f(x) = x$ makes $f$ continuous at $0$). So how the preceding definition can be fixed in order to include all definable functions $f: I \subset \mathbb{R} \rightarrow \mathbb{R}$ ?