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.
accumulation point of A, and the condition for f to be continuous at 0 is that lim n!1 yn = y0. As for limits, we can give an equivalent sequential definition of continuity, which follows immediately from Theorem 2.4. Theorem 3.6. If f: A → R and c ∈ A is an accumulation point of A, then f is continuous at c if and only if lim n!1 f(xn) = f(c)
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