About the product of two uniformly continuous functions

Question

I am interested in the following question:

Assume $f,g: [0,+\infty)\to\mathbf R$ are both uniformly continuous, and $g(x)\to 0$ as $x\to+\infty$. Can this guarantee that $h(x)\equiv f(x)g(x)$ is uniformly continuous on $[0,+\infty)$?

Any hint is highly appreciated!

Answer 1

This question does not need an "answer" since the example given by @RyszardSzwarc completes the assignment most elegantly.

But that never stops anyone with a particularly strong pedantic inclination. I learned from L.C. Young who was a professor at Madison and the son of a pair of famous British mathematicians of something he called "lemon squeezing" in one of his memoirs.

Given a problem or a theorem, a mathematician is expected to gnaw and chew at it like a dog on an old bone. Or, to use his much better metaphor, squeeze it for just a wee bit more, similar in the way one can always get just one more drop of lemon juice from a lemon if you try a bit harder.

So we know that the product of two uniformly continuous functions on a compact interval $[a,b]$ must be uniformly continuous, but that the product of such functions on the unbounded interval $[0,\infty)$ need not be. Even if one of the pair has a zero limit at $\infty$. End of assignment. Time for a beer.

Or, maybe, squeeze that lemon. If you wish to transition from a student to a research mathematician then you really must always get used to doing that lemon squeezing...along with the beer if you must.

Problem I. Give necessary and sufficient conditions on a function $h:[1,\infty)\to\mathbb R$ so that it can be written as a product $h(x)=f(x)g(x)$ where both functions $f$ and $g$ are uniformly continuous on $[1,\infty)$.

Problem II. Give necessary and sufficient conditions on a function $h:[1,\infty)\to\mathbb R$ so that it can be written as a product $h(x)=f(x)g(x)$ where (i) both functions $f$ and $g$ are uniformly continuous on $[1,\infty)$, and (ii) $\lim_{x\to\infty}g(x) = 0$.

For Problem I you might wish to consult this reference:

On the Product of two Uniformly Continuous Functions on the Line. Ernest S. Elyash, George Laush and Norman Levine. The American Mathematical Monthly, Vol. 67, No. 3 (Mar., 1960), pp. 265-267 https://doi.org/10.2307/2309692