Problem 5 Show that if f(x) is bounded on [a, b] and continuous on [a, b]\E, where E is a finite set, then f R[a, b]. Also show that you can not remove the boundness assumption.

* Suppose that f: R R is continuous. Show that, for any c E R, the function -> cf : R Ris also continuous. (cf is the function obtained by multiplying the output of f by c.)

Suppose that the function f: R → Ris continuous and that its image f (R) is bounded. Prove that there is a solution of the equationf(x) = x, x in R.

3. Let f: R → R be a continuous function. Show that the set K = {x|f(x)=0} is a closed set.

3. Let S {1,2, 3, 4} and define functions f,g : S → S by f = {(1,3), (2,2), (3, 4), (4, 1)} and g = {(1, 4), (2, 3), (3, 1), (4, 2)}. Find (a) g-1 o f og (b) f oglog (c) f-1 og –1 -1ofog