A Second Course on Real Functions
When considering a mathematical theorem one ought not only to know how to prove it but also why and whether any given conditions are necessary. All too often little attention is paid to to this side of the theory and in writing this account of the theory of real functions the authors hope to rectify matters. They have put the classical theory of real functions in a modern setting and in so doing have made the mathematical reasoning rigorous and explored the theory in much greater depth than is customary. The subject matter is essentially the same as that of ordinary calculus course and the techniques used are elementary (no topology, measure theory or functional analysis). Thus anyone who is acquainted with elementary calculus and wishes to deepen their knowledge should read this.
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absolutely continuous analytic sets antiderivative bijection Borel function Borel measurable Borel set bounded variation Cantor function Choose class of Baire closed interval closed set closed subset constant construct contains continuous surjection converges convex COROLLARY countable set Darboux continuous functions definition denote differentiable a.e. differentiable function Dini derivatives everywhere example f is continuous F„-sets finite function g functions R-ğR graph Hence Hint increasing function indefinite integral initial interval intersection L-set Lebesgue integrable Lebesgue measurable set Lemma Let f Let g lower semicontinuous Math meagre set monotone functions nonempty Notes to Section null set obtain open interval open set pairwise disjoint Perron integrable points of discontinuity real number restriction Riemann integrable Riemann integrable function Riemann sequence semicontinuous function Show smallest element strictly increasing subinterval surjection union of countably well-ordered set x e(a