This is Analysis.

Please prove every part clearly.

a domain D in R. Problem 4. Write the definition and the negation of the definition of uniform continuity for a function f on (i) Prove that if f is uniformly continuous on D and {an} is a Cauchy sequence in D then {f(an) } is a Cauchy sequence. (ii) Give an example of a continuous function f : (0, 1) - R and a Cauchy sequence {an}."? 1 in (0, 1) for which {f (an) } 1 is not Cauchy. Can this happen if f is continuous in D = [0, 1]? (iii) Prove that the function x - - is not uniformly continuous on D = R \\ {0}. Hint: you can either use the negation of the definition or part (iv) If f is continuous on D = [a, b], and f + 0 on D, prove that the function = is uniformly continuous on D in two ways: (a) Use the definition. Hint: write f(x) f(2) If (y) - f (x)l If (x)f (y)l and bound |f(x)f (y)| from below. (b) Use a theorem