some uniform continuous functions

We need to find which are uniform continuous (UC) on a) $(0,1)$ and b) $(0,\infty)$. I have done, could you confirm me, if I am wrong any where?

1. $\frac{1}{(1-x)}$

2. $\frac{1}{(2-x)}$

3. $\sin x$

4. $\sin(1/x)$

5. $x^{1/2}$

6. $x^3$

1) is not UC on a) because limit does not exist when $x\rightarrow 1$, on b) it is discontinous (disco) at $x=1$

2) is UC on a) and it is disco at $x=2$ so not UC on b)

3) is UC on a) and also on b) as we can show by the inequality $|\sin x-\sin y|<|x-y|$

4) is not UC on a) as limit does not exist, and also not UC on b) I am not clear enough for this one.

5) is UC on a) and not UC on b) as derivative is not bounded near $0$

6) is UC on a) and not UC on b) as derivative is not bounded.