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2 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all xX z y x z y x z (A) (B) (C) FIGURE 8 (i) x y (ii) x y (iii) x y FIGURE 9 solution The projection of curve (C) onto the xyplane is neither a segment nor a periodic wave Hence, the correct projection is (iii), rather than the two other graphs The projection of curve (A) onto the xyplane is a vertical line, hence the corresponding projection@ a b c d e f g h i j k l m n o p q r s t u s v w @ k r g?

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