Transcription of Chapter 3
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Chapter 3
Chapter3 Continuous FunctionsIn this Chapter , we define continuous functions and study their ContinuityAccording to the definition introduced by Cauchy, and developed by Weierstrass,continuous functions are functions that take nearby values at nearby nition :A R, whereA R, and suppose thatc A. Thenfiscontinuous atcif for every >0 there exists a >0 such that|x c|< andx Aimplies that|f(x) f(c)|< .A functionf:A Ris continuous on a setB Aif it is continuous at everypoint inB, and continuous if it is continuous at every point of its definition of continuity at a point may be stated in terms of neighborhoodsas nition functionf:A R, whereA R, is continuous atc Aif forevery neighborhoodVoff(c) there is a neighborhoodUofcsuch thatx A Uimplies thatf(x) V .The - definition corresponds to the case whenVis an -neighborhood off(c)andUis a -neighborhood ofc. We leave it as an exercise to prove that thesedefinitions are thatcmust belong to the domainAoffin order to define the continuityoffatc.
continuous on R. Repeated application of Theorem 3.15 for scalar multiples, sums, and products implies that every polynomial is continuous on R. It also follows that a rational function R = P/Q is continuous at every point where Q ̸= 0.
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