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on $[0,\infty)$, can we prove $f(x)g(x)$ is uniformly continuous if $f,g$ are uniformly continuous and $\lim_{x\to \infty}g(x)=0$?

If $\lim_{x\to \infty}g(x)=0$ is droped, then it is easy to see that it is wrong with the example $f(x)=g(x)=x$.

xldd
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  • Actually the question should be, Prove that if f and g are bounded and uniformly continuous on an interval. Then the product fg is also uniformly continuous at that interval. And also bounded is not necessary for uniform continuity as you do in your first assumption. But in your second assumption boundedness is necessary. – Unknown Nov 16 '18 at 01:54

1 Answers1

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Let $f(x)=x$ and $g(x)=\frac {\sin(x^{2})} x$ so that $f(x)g(x)=\sin(x^{2})$. $g$ is uniformly continuous because $g'$ is bounded. Also $g(x) \to 0$ as $x \to \infty$. Now let $x_n=\sqrt {n\pi}$ and $y_n=\sqrt {(n+\frac 12 ) \pi}$. Then $\sin (x_n^{2})-\sin (y_n)^{2}=0-1=-1$ for all $n$ even though $|x_n-y_n|=\frac {\frac 1 2 \pi} {x_n+y_n} \to 0$ as $n \to \infty$. Hence $fg$ is not uniformly continuous.

Kavi Rama Murthy
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