Pointwise product of uniformly continuous functions

Question

True or false: Let $f(x)$ and $g(x)$ be uniformly continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Then their pointwise product $f(x)g(x)$ is uniformly continuous.

I think it will be true; take $f(x)= g(x) = \sqrt x$.

Am I right or wrong?

Answer 1

It is false. Just take $f(x)=g(x)=x$.

Answer 2

The uniform continuity property refers to a spatially homogeneous behavior of continuity. Look at the graph of the following functions.

1. $f(x)=x$
2. $f(x)=\sqrt{|x|}$
3. $f(x)=x^2$
4. $$ f(x)= \begin{cases} x\sin(\frac{1}{x}) & x\ne 0 \\ 0 & x=0 \end{cases}$$

Note that (2) has a "vertiginous behavior" around $0$, since that $$f'(x)=\frac{1}{2\sqrt{x}}\to \infty$$ when $x\downarrow 0$. Also for (3) this behavior occurs when $|x|\to \infty$. (4) is continuous but is even "less uniform" than previous cases.