Intro to Analysis in-class_Part_32

Intro to Analysis in-class_Part_32 - 4.2 Properties of...

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4.2 Properties of continuity 69 Since f is monotonic, f ( x - δ ) f ( t ) A for x - δ < t < x. Combine prev two eqns to get | f ( t ) - A | < ε . Thus, A = f ( x - ). Next, for a < x < y < b , monotonicity gives sup a<t<y f ( t ) = sup y - δ<t<y f ( t ) = f ( y - ) from which f ( x +) = inf a<t<y f ( t ) sup a<t<y f ( t ) = f ( y - ) . Corollary 4.2.11. A monotone function on an interval has at most a countable number of discontinuities, all of which are jump discontinuities. Proof. Wlog, let f be increasing, and let D be the set of discontinuities of f . For each x D , associate a rational number r ( x ) such that f ( x - ) < r ( x ) < f ( x +). Since x < y = f ( x +) f ( y - ) , the strict inequalities above give x 6 = y = r ( x ) 6 = r ( y ) . r is injective, and thus gives a bijection between the jumps and a subset of Q . § 4 Exercise: #2,5,11 Recommended: #3,9 1. Show f is continuous iff the preimage of every closed set is closed. 2. Show that a continuous function is determined by its values on a dense subset of the
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_32 - 4.2 Properties of...

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