Let $g$ be a function from the hyperreal numbers to the hyperreal numbers which is an extension of a real function in the canonical way, satisfying the condition that for all real numbers $x$ and all positive infinitesimals $\epsilon$, $g(x+\epsilon)=g(x)+\epsilon$. Then my question is, what condition does the “standard part” of $g$ (i.e. considered as a function from $\mathbb{R}$ to $\mathbb{R}$) satisfy?
Note that the word positive is important; I don’t want it to necessarily hold for negative infinitesimals. In any case, this is all in service of my question here.
By transfer, if $a\in\mathbb{R}$ and $g(a+\epsilon)=g(a)+\epsilon$ is true for all infinitesimal $\epsilon>0$, it must also be true for all sufficiently small real $\epsilon>0$. So your condition is equivalent to the condition (in the following all variables are real) that for each $x$ there exists $\delta>0$ such that $g(x+\epsilon)=g(x)+\epsilon$ for all $0\leq\epsilon<\delta$. Or, letting $h(x)=g(x)-x$, this means that $h$ is constant in a half-open interval starting at each point. (So, $\mathbb{R}$ can be partitioned into half-open intervals $[a,b)$ such that $h$ is constant on each one.)
If you removed the positivity condition on $\epsilon$, you would similarly conclude that $h$ is constant in an open interval around each point and thus globally constant since $\mathbb{R}$ is connected, so $g(x)$ would have the form $x+c$ for some constant $c$.