1. Suppose that f : [a, b] → and g : [a, b] → are continuous functions such that f (a) ≤ g (a) and f (b) ≥ g (b). Prove that f (c) = g (c) for some c ∈ [a, b]. 2. Suppose f : [a, b] → is continuous...

1. Suppose that f : [a, b] →

and g : [a, b] →

are continuous functions such that f (a) ≤ g (a) and f (b) ≥ g (b). Prove that f (c) = g (c) for some c ∈ [a, b].

2. Suppose f : [a, b] →

is continuous and that f ([a, b]) ⊆
. Prove that f is constant on [a, b].

3. Suppose that f : [a, b] →

is two-to-one. That is, for each y ∈
, f^–1
({ y}) either is empty or contains exactly two points.

(a) Find an example of such a function.

(b) Prove that no such function can be continuous