Examples of continuous but not uniformly continuous functions on various domains

Question

What would be some simple examples of functions that are continuous but not uniformly continuous on the following intervals?

1. $D=\mathbb R$
2. $D=(0,1]$
3. $D=[0,1]$

This is what I came up with:

1. $f(x)=1/\sin(x)$
2. $f(x)=1/x$
3. $f(x)=?$

Are these correct? Could it be that the third case is impossible? If so, why?

Thanks to anyone for the help!!