If $f(x)$ is continuously decreasing and $\lim_{x \to \infty} f(x) = 0$, is $xf(x)$ uniformly continuous?

Choose a strictly increasing sequence of integers $0=a_0<a_1<\cdots$. Now define $f$ as follows. If $a_k\leq x\leq a_{k+1}-1$ for some $k$ then $f(x)=2^{-k}$; on the interval $[a_{k+1}-1,a_{k+1}]$ the function is linear.

Now on the interval $(a_{k+1}-1,a_{k+1})$ we have $g(x)$ is differentiable with derivative at most $2^{-k}-2^{-k-1}(a_{k+1}-1)$; call this bound $b_k$. If we choose $a_k$ increasing sufficiently quickly, $b_k\to-\infty$. Thus for $x_k$ in the middle of this interval and any given $\varepsilon>0$ we need $\delta\leq\varepsilon/|b_k|$, so no fixed $\delta>0$ works.