A function $f$ on an interval $I$ is uniformly continuous if its natural extension $f^{*}$ in $I^{*}$ has the following property: > > for every pair of hyperreals $x$ and $y$ in $I^{*}$, if $x\approx y$ then $f^*(x)\approx f^*(y)$

Question is, for the first "if": Can this be changed into "if and only if"?

The same question for the quote below:

A real function $f$ is continuous at a standard real number $x$ if for every hyperreal $x'$ infinitely close to $x$, the value $f(x')$ is also infinitely close to $f(x)$. This captures Cauchy's definition of continuity.


Yes. If $f$ is uniformly continuous and $x,y \in I^*$ with $x \approx y$ are given, then for fixed $\varepsilon \in \mathbb{R}_{> 0}$ you can choose $\delta \in \mathbb{R}_{> 0}$ such that for all $r,s \in I$ such that $|r-s| < \delta$ we have $|f(r)-f(s)| < \varepsilon$. By transfer this means $$\forall\, r,s \in I^* \: : \: |r-s| < \delta^* \Rightarrow |f^*(r)-f^*(s)| < \varepsilon^*$$ But $|x-y| < \delta^*$, so we have $|f^*(x)-f^*(y)| < \varepsilon^*$. Since $\varepsilon$ was arbitrary, $f^*(x) \approx f^*(y)$.

The same reasoning can be applied for pointwise continuity.


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