5. Choose whether each of the following statements are true or false. (a) If f is a real-valued continuous function on the closed interval [a; b], then f is uniformly continuous. (b) Let (Mi ; di ) metric spaces for i = 1; 2 and f : M1 → M2 be continuous. Then f(U) is open in M2 for all open subset U in M1. (c) Every homeomorphism is uniformly continuous.
5. Choose whether each of the following statements are true or false.
(a) If f is a real-valued continuous function on the closed interval [a; b], then f is uniformly continuous.
(b) Let (Mi ; di ) metric spaces for i = 1; 2 and f : M1 → M2 be continuous. Then f(U) is open in M2 for all open subset U in M1.
(c) Every homeomorphism is uniformly continuous.
(d) Every contraction mapping has a fixed point.
(e) An identity function on any metric space is a contraction.
(f) A surjective isometry is a homeomorphism.
(g) If f , g : R→R are uniformly continuous function on R then the product f * g : R→ R is uniformly continuous on R.
For a) we use the theorem any continuous function on a compact set is uniformly continuous.
For b) we find a counter example
For c) also we find a counter example
Note: According to our guidelines we can solve only first three subpart and rest can be reposted .Thank you
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