1. one more topology question

(S,d) and (S*,d*) are metric spaces, f:S->S* continuous functions, E subset of S are given.

i answered part 1, which was to show that E is path-connected implies f(E) path-connected. this may help solve part 2 and 3...
here are parts 2 and 3 of the question:
2. (note R denotes the set of reals) let f:[a,b] -> R, then there exists x' and x" in [a,b] such that f(x')=inf f(x) where x is in [a,b] and f"=sup f(x) where x is in [a,b]

3. for every pt y in [f(x'),f(x")], there exists a point x in [a,b] such that f(x)=y. hint given by instructor is to consider the compactness and path-connectedness of [a,b], and that this is a stronger version of the intermediate value thm.

can i get some hints on how to approach parts 2 and 3? thanks.

2. Originally Posted by squarerootof2
(S,d) and (S*,d*) are metric spaces, f:S->S* continuous functions, E subset of S are given. i answered part 1, which was to show that E is path-connected implies f(E) path-connected. this may help solve part 2 and 3...
here are parts 2 and 3 of the question:
2. (note R denotes the set of reals) let f:[a,b] -> R, then there exists x' and x" in [a,b] such that f(x')=inf f(x) where x is in [a,b] and f"=sup f(x) where x is in [a,b]
3. for every pt y in [f(x'),f(x")], there exists a point x in [a,b] such that f(x)=y. hint given by instructor is to consider the compactness and path-connectedness of [a,b], and that this is a stronger version of the intermediate value thm.can i get some hints on how to approach parts 2 and 3? thanks.
It is confusing to change function notation within one problem.
It seems that part 2 is simply asking you to prove the classic “High/Low point” theorem for any real value continuous function on a compact set: On a closed interval a continuous function has a maximum and a minimum.