Let $f:[0,\infty)\to (0,\infty)$ be a continuous function and $\displaystyle\int^{\infty}_{0}f&lt;\infty$. Show that $f$ is uniform continuous iff $\displaystyle\lim_{x\to \infty}f(x)=0$   So I could only see that continuity is not sufficient condition for the limit to exists since $\displaystyle\int^{\infty}_{0} \cos x^2$ exists but $\displaystyle\lim_{x\to \infty}\cos x^2$ does not exist , so we need something more stronger than continuity.
Trying to solve     $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx  $ converges.   I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$  Would appreciate your help!
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