If $f,g$ are uniformly continuous prove $f+g$ is uniformly continuous but $fg$ and $\dfrac{f}{g}$ are not

Solution 1:

A product of uniformly continuous functions is not necessarily uniformly continuous. For example, set $E=\mathbb{R}$ and choose $f(x)=g(x)=x$. Their product, $x^2$, is an example of a nonuniformlycontinuous function.

Solution 2:

Product $fg$

Boundnness is suffient but not necessary for uniform continuity of product $fg$.

Here is an example where $f,g$ both are unbounded but their product is uniformly continuous: Let $E=[0,\infty)$ and $f(x)=g(x)=\sqrt{x}$. Here $f$ and $g$ are unbounded but their product $(fg)(x)=x$ is uniformly continuous.

$\frac fg$:
It is not true that $\frac fg$ is always uniformly continuous. Let $E=[1,\infty)$,$f(x)=x$ and $g(x)=\frac{1}{x}$ then $\left(\frac fg\right)(x)=x^2$ is not uniformly continuous on $[1,\infty)$.