Give an example of each of the following or state that such

Give an example of each of the following, or state that such a
request is impossible. For any that are impossible, supply a short explanation
for why this is the case.
(a) A continuous function f : (0, 1) R and a Cauchy sequence (xn) such that f(x_n) is not a Cauchy sequence;
(b) A uniformly continuous function f : (0, 1) R and a Cauchy sequence (x_n) such that f(x_n) is not a Cauchy sequence;
(c) A continuous function f : [0,) R and a Cauchy sequence (x_n) such that f(x_n) is not a Cauchy sequence;

Solution