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
(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.

2. Originally Posted by Swamifez
(b) There exists a function f : [−1, 1] → [−1, 1] with no x ∈ [−1, 1]
satisfying f (x) = x.
It depends whether f is continous or not. If it is then there is no such function, this is a "Brouwer Fixed Point (I think)"

3. The Perfect Hacker, thank you for your post before. assume f is contimous in all my problems. Thanks again

4. Originally Posted by Swamifez
1.Decide whether the following statements are true or false. Justify your
The limit $\lim_{x\to 1^-}f(x)$ exists for $f$ is continous and bounded. Let $L$ be that limit.
$g(x)=\left\{ \begin{array}{c}f(x) \mbox{ for }0\leq x<1\\ L \mbox{ for }x=1 \end{array}\right\}$
This new function is countinous on the closed interval $[0,1]$ EVT tells us it has a maximum value.