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.


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