Appendix A

The Theory Of The RiemannnIntegral∗

A.1 An Important WarningIf you read and understand this appendix on the Riemann integral you will become abnor-mal if you are not already that way. You will laugh at atrocious puns. You will be unpopularwith well adjusted confident people, especially religious people who love to accept on faithinconsistent decrees of authority figures. Furthermore, your confidence will be completelyshattered. Virtually nothing will be obvious to you ever again. Consider whether it wouldbe better to accept the superficial presentation given earlier than to attempt to acquire deepunderstanding of the integral, risking your self esteem and confidence, before proceedingfurther. This is only here for those who need explanations and are not content to accept onfaith. This chapter is one of the worst things I have seen and I don’t know how to improveit without losing the rigor. I think it is a good illustration why, if you want to do integra-tion, you should approach it as a Lebesgue integral. This is found in my book Calculus offunctions of real and complex variables or Calculus of One and Many Variables. It will bemuch more abstract, but much less filled with mind numbing technicalities.

A.2 Basic DefinitionThe definition of the Riemannn integral of a function of n variables uses the followingdefinition.

Definition A.2.1 For i = 1, · · · ,n, let{

α ik

}∞

k=−∞be points on R which satisfy

limk→∞

αik = ∞, lim

k→−∞α

ik =−∞, α

ik < α

ik+1. (1.1)

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