# Example of an internal function which is $\epsilon - \delta - continuous$ but not $s-continuous$

Problem: I was looking for a function which is $$\epsilon - \delta - continuous$$ but is not $$s-continuous$$ at some point.

Here are the definitions :

$$s-continuous$$ : An internal function $$f \subset^* \mathbb{R} \times \mathbb{R}$$ is called $$s-continuous$$ at a point $$x \in \mathbb{R}$$,, then its *-extension $$^*f$$ is $$s-continuous$$ at the point $$x$$.

$$\epsilon - \delta-continuous$$ : An internal function $$f \subset^* \mathbb{R} \times ^*\mathbb{R}$$ is called $$\epsilon - \delta-continuous$$ at a point $$x_0 \in ^* \mathbb{R}$$ if for any $$\epsilon \in ^* \mathbb{R}^+$$, then there exists a $$\delta \in ^* \mathbb{R}^+$$ , such that $$(|x-x_0|< \delta) \implies (|f(x) - f(x_0)| < \epsilon)$$.

I am thinking about the function $$f(x) = x^2$$ which is a $$\delta - \epsilon - continuous$$ but not sure about $$s-continuity$$. Hoping for your help.