First example: (f is continuous at a) iff (x ≈ a implies f(x) ≈ f(a)), where a ≈ b iff a–b is infinitesimal. Your definition of continuity within *R is the same as mine within †R, both simpler (and much more intuitive!) than the standard definition within R: (f is continuous at a) iff...
All I am doing, is trying to establish the relative consistency of an ordered field of hyperreal numbers, †R, with that of its subfield, the Dedekind complete set of standard real numbers, R. To this end, I am using the non-Archimedean ordered field real rational functions in a single variable...
f(n) = n is a rational function of n, so that f is an element of †R. It is also true that √f is not. You could call √f an irrational function, that it is an element of *R–†R.
But, just as one may approximate √2 to any desired precision with rational numbers, so one may approximate √f to any...
All I am trying to do is to show that making dx an "infinitely small bit of x" (to paraphrase Thompson's Calculus Made Easy) does not lead to any contradictions with any of the first order axioms that define the real numbers.
Robinson did this with equivalent sets of all possible real...
Calculus Made Easy by Sylvanus P. Thompson has been an excellent introduction the the subject, suitable for high school or first year college students for a hundred years. The 1910 edition, now in the public domain, has some obsolescent terminology and notation Martin Gardner updated in the...
The usual approach to nonstandard (or infinitesimal) calculus (based on Abraham Robinson's work, Non-standard Analysis, 1962) does indeed depend on ultrafilters and the Axiom of Choice, and is usually reserved for third year university level or even post graduate courses. This is a shame...