For the following question: if true, prove it; if false, give a counter-example.

1. Assume $f: X \to Y$ is uniformly continuous, and $g: Y \to Z$ is uniformly continuous. Is it true that $g \circ f: X \to Z$ is uniformly continuous?

2. If $f: (0,1) \to (0,1)$ is a continuous map, is it true that there exists a $x \in (0,1)$, such that $f(x) = x$? What if one change the setting to $f: [0,1] \to [0,1]$ being continuous, does your conclusion change?

3. A function $f: \R \to \R$ is called a Lipschitz function, if there exists a $M>0$, such that $|f(x)-f(y)| / |x-y| \leq M$ for any $x \neq y \in \R$. Is it true that Lipschitz functions are uniformly continuous? Is it true that all uniformly continuous functions are Lipschitz?

4. Consider the following function $f: (0,1) \to \R$, $f(x) = x \sin (1/x)$. If $f$ uniformly continuous? Give a proof for your claim.

5. Is there a monotone function on $[0,1]$, such that it is discontinuous at $1/2, 1/3, 1/4, \cdots$? If so, give a construction; if no, give a proof.