Encouraged by this question I ask a question concerning Keisler's treatment of definite integrals using infinitesimals.
He starts with a continuous function $f$ defined on a closed interval $[a, b] $ and forms an infinite Riemann sum $\sum_{a} ^{b} f(x) \, dx$ where the infinitesimal $dx$ is obtained by partitioning the interval $[a, b] $ into $H$ parts ($H$ being a hyperinteger) and $dx=(b-a) /H$.
Next he uses the fact that $f$ has a minimum value and maximum value so that it is bounded and then proves that the infinite Riemann sum $\sum_{a} ^{b} f(x) \, dx$ is a finite hyperreal number and we can take its standard part to get a real number which is defined as the integral $\int_{a} ^{b} f(x) \, dx$.
But looking at the overall definition I don't see continuity being used in essential manner apart from what is mentioned in last paragraph. Thus the argument applies equally to any bounded function on $[a, b] $ and it's integral is defined. But as we know from standard analysis this is not the case. Not all bounded functions are Riemann integrable, but continuous functions are. Thus it appears that the overall presentation skips the important fact that continuity implies integrability. Most introductory calculus texts instead mention this fact without proof.
My question is:
Is the presentation of Riemann integral by Keisler rigorous enough or a bit hand wavy as previous paragraph indicates? And if it is hand wavy, how can the presentation be improved using infinitesimal approach to include the proof that continuity implies integrability?
Keisler's book is available for download here and the definite integrals are discussed in chapter 4.
There is no handwaving here.
It is true that not every bounded real function on an interval is Riemann integrable, but that is not needed here. The argument does indeed show that all infinite Riemann sums for a bounded function are well defined.
Note that Keisler starts by defining "the definite integral of $f$ from $a$ to $b$ with respect to $dx$". The "with respect to"-part matters. The point is that for an arbitrary bounded function, the definite integral with respect to an infinitesimal can depend on the infinitesimal.
Then in Theorem 1 of 4.2., Keisler shows that for $f$ continuous, the choice of the infinitesimal does not matter. This is exactly where continuity is needed.