The definition of a derivative seems wrong. For example, suppose that f(x)=0 for rational x but f(x)=1 for irrational x. Then f is not differentiable anywhere, but according to your definition it would have a derivative of 0 everywhere (since Δx could be an infinitesimal consisting of a sequence of only rational numbers).
Have updated the definition of the derivative to specify the differences between f over the hyperreals and f over the reals.
I think the natural way to extend your f to the hyperreals is for it to take values in an infinitesimal neighborhood surrounding rationals to 0 and all other values to 1. Using this, the derivative is in fact undefined, as st(0/Δx)=0/st(Δx)=0/0.
The definition of a derivative seems wrong. For example, suppose that f(x)=0 for rational x but f(x)=1 for irrational x. Then f is not differentiable anywhere, but according to your definition it would have a derivative of 0 everywhere (since Δx could be an infinitesimal consisting of a sequence of only rational numbers).
Have updated the definition of the derivative to specify the differences between f over the hyperreals and f over the reals.
I think the natural way to extend your f to the hyperreals is for it to take values in an infinitesimal neighborhood surrounding rationals to 0 and all other values to 1. Using this, the derivative is in fact undefined, as st(0/Δx)=0/st(Δx)=0/0.