Products of unbounded uniformly continuous functions
= Products of unbounded uniformly continuous functions
A product of two <uniformly continuous> functions need not have <uniform continuity> on an unbounded domain. The functions $f(x)=g(x)=x$ give the counterexample $x^2$, since inputs $n$ and $n+1/n$ approach each other while their squared values remain separated. For bounded factors the estimate $|f(x)g(x)-f(y)g(y)|\le\|f\|_\infty|g(x)-g(y)|+\|g\|_\infty|f(x)-f(y)|$ restores uniform continuity.