By Bernard R. Gelbaum

These counterexamples, prepared in accordance with hassle or sophistication, deal in general with the a part of research often called "real variables," beginning on the point of calculus. the 1st 1/2 the publication matters services of a true variable; subject matters comprise the true quantity method, capabilities and boundaries, differentiation, Riemann integration, sequences, endless sequence, uniform convergence, and units and degree at the genuine axis. the second one part, encompassing greater dimensions, examines services of 2 variables, aircraft units, zone, metric and topological areas, and serve as areas. This quantity includes a lot that might turn out appropriate for college kids who've no longer but accomplished a primary path in calculus, and considerable fabric of curiosity to extra complicated scholars of study in addition to graduate scholars. 12 figures. Bibliography. Index. Errata.

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A in the neighborhood homogeneous always differentiable functionality of 2 variables that isn't homogeneous thirteen. A differentiable functionality of 2 variables owning no extremum on the foundation yet for which the restrict to an arbitrary line in the course of the starting place has a strict relative minimal there 14. A refinement of the previous instance 15. A functionality f for which even supposing every one necessary is right sixteen. A functionality f for which even if every one critical is right 17. A double sequence for which even supposing convergence holds all through 18. A differential P dx + Q dy and a airplane area R during which P dx + Q dy is in the community designated yet now not targeted 19. A solenoidal vector box outlined in a simply-connected quarter and owning no vector strength 10. airplane units advent 1. disjoint closed units which are at a nil distance 2. A bounded aircraft set contained in no minimal closed disk three. “Thin” attached units that aren't basic arcs four. disjoint aircraft circuits contained in a sq. and connecting either pairs of contrary vertices five. A mapping of the period [0, 1] onto the sq. [0, 1] × [0, 1] 6. A space-filling arc within the airplane 7. A space-filling arc that's nearly all over inside a countable set eight. A space-filling arc that's nearly far and wide differentiable nine. a continuing mapping of [0, 1] onto [0, 1] that assumes each worth an uncountable variety of occasions 10. an easy arc within the unit sq. and of aircraft degree arbitarily close to 1 eleven. A hooked up compact set that's not an arc 12. A aircraft zone diversified from the internal of its closure thirteen. 3 disjoint airplane areas with a standard frontier 14. A non-Jordan sector equivalent to the inner of its closure 15. A bounded airplane quarter whose frontier has optimistic degree sixteen. an easy arc of endless size 17. an easy arc of countless size and having a tangent line at each element 18. an easy arc that's of countless size among each pair of distinctive issues at the arc 19. A tender curve C containing some degree P that's by no means the closest element of C to any aspect at the concave part of C 20. A subset A of the unit sq. S = [0, 1] × [0, 1] that's dense in S and such that each vertical or horizontal line that meets S meets A in just one element 21. A nonmeasurable airplane set having at such a lot issues in universal with any line 22. A nonnegative functionality of 2 variables f(x, y) such that and such that the place S = [0, 1] × [0, 1], doesn't exist 23. A real-valued functionality of 1 genuine variable whose graph is a nonmeasurable airplane set 24. A hooked up set that turns into completely disconnected upon the removing of a unmarried element eleven. zone advent 1. A bounded airplane set with out sector 2. A compact aircraft set with no sector three. A bounded aircraft area with no quarter four. A bounded airplane Jordan quarter with out region five. an easy closed curve whose airplane degree is bigger than that of the bounded zone that it encloses 6. services φ and ψ outlined on [0, 1] and such that (a)φ(x) < ψ(x) for ; (b) exists and is the same as 1; (c)S ≡ {(x, y) 0x1, φ(x)

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