Define what is meant by a uniformly continuous function on a set $E \subset \mathbb{R}$.

If $f$ and $g$ are uniformly continuous functions on $\mathbb{R}$, is the (pointwise) product $f g$ necessarily uniformly continuous on $\mathbb{R}$ ?

Is a uniformly continuous function on $(0,1)$ necessarily bounded?

Is $\cos (1 / x)$ uniformly continuous on $(0,1) ?$

Justify your answers.