Dear math helpers, thank you for all your help thus far. These are two more problems I was having trouble with and would appreciate feedback. Thank you in advance
1.Decide whether the following statements are true or false. Justify your
answers: proof or counterexample.
(a) Every continuous function f : [0, 1) → R which is bounded
takes on its maximum.
(b) There exists a function f : [−1, 1] → [−1, 1] with no x ∈ [−1, 1]
satisfying f (x) = x.
2.Suppose that the function f is diﬀerentiable at a. Prove (without
quoting a theorem) that f 2 is diﬀerentiable at a.