2 more fun problems, please help!

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.