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.
In fact, taking a rational sequence (xn) and an irrational sequence (˜xn) that converge to c, we see that limx!c f(x) does not exist for any c ∈ R. Example 3.14. The Thomae function f: R → R defined by f(x) = {1/q if x = p/q where p and q > 0 are relatively prime, 0 if x /∈ Q or x = 0 is continuous at 0 and every irrational number and ...
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