# real analysis

S SOLUTION: real analysis home X S 20210418201505 2021_0 X YouTube х New Tab X + studypool.com/uploads/questions/817105/20210418201505 _2021_04_19_04.14.36.png C Paused Update : !! Apps N Netflix Stream TV and M… ISU portal Q Quizlet Bb Blackboard Learn p Pinterest S Question Feed – S… O Mail – Cheyenne H… r Final Grade Calcul… Indiana State Univ… EXERCISES 4.3 1. Prove Theorem 4.3.3. 2. Show that the following functions are not uniformly continuous on the given domain. *a. f(x) = x. Dom f = (0.00) b. 8(x) = c. h(x) = sin 3. Prove that each of the following functions is uniformly continuous on the indicated set. *a. f(x) – Tx: +€ (0.00) b. 8(x) = x, XEN – Domg = (0.00) sin Dom h = (0.00) XER d. k(x) = cos x, XER c. h(x) = 2 6. e(x) = sinx € (0.00) *. f(x) = € (0, 0) 4. Show that each of the following functions is a Lipschitz function. •2. f(x) – Dom f = [0,00), a > 0 b. 8(x) = Dij Dom 8 = [0,00) ch(x) = sin sin Dom h = [0,00), a > 0 d. p(x) a polynomial, Domp = (-a, a), a > 0 5. a. Show that f(x) = Vx satisfies a Lipschitz condition on (a,00), a > 0. b. Prove that is uniformly continuous on (0.00). c. Show that does not satisfy a Lipschitz condition on (0,0). 6. Suppose ECR and f. 8 are Lipschitz functions on E. a. Prove that f + g is a Lipschitz function on E. b. If in addition f and g are bounded on E, or the set E is compact, prove that fg is a Lipschitz function on E. 7. Suppose ECR and f. 8 are uniformly continuous real-valued functions on E. a. Prove that f + g is uniformly continuous on E. *b. If, in addition, and g are bounded, prove that fg is uniformly continuous on E. c. Is part (b) still true if only one of the two functions is bounded? 8. Suppose ECR and f: ER is uniformly continuous. If {xa} is a Cauchy sequence in E, prove that {f(x)} is a Cauchy sequence 9. Let f:(a,b)-R be uniformly continuous on (a, b). Use the previous exercise to show that f can be defined at a and b such that f is continuous on (a, b). 10. Suppose that E is a bounded subset of R and f: ER is uniformly continuous on E. Prove that f is bounded on E.

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