Let a, b ∈ R with a < b and let f : [a, b] → [a, b] be continuous. Show that there exists x0 ∈ [a, b] such that f(x0) = x0.

Show that if f : [a, b] → R is continuous and ) x a f = 0 for all x ∈ [a, b], then f(x) = 0 everywhere on [a, b]. Provide an example to show that this conclusion does not follow if f is not continuous.

Determine whether the following statements are true or false. If true, provide a proof; if false, provide a counterexample. (a) Let f, g, h be continuous on the interval [a, b]. If f(a) g(b) > h(b), then there exists c = [a, b] such that f(c) = g(c) = h(c). (b) Suppose that f and g are continuous on R. If 0 ≤ f(x) < g(x) for all X, then there is some x ER such that f(x)/g(x) is the maximum value of f/g. (c) If f is continuous on R, then f is bounded.

Let f, g : [a, b] → [a, b] be two continuous functions that satisfy f ◦ g =g ◦ f. Show that there exists x0∈ [a, b] such that f(x0) = g(x0).