Barbălat’s lemma

Lemma (Barbălat).

Let $f\mathrm{:}\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{\infty }\mathrm{)}\mathrm{\to }\mathrm{R}$ be Riemann integrable and uniformly continuous then

$$\underset{t\to \mathrm{\infty }}{lim}f(t)=0.$$

Note that if $f$ is non-negative, then Riemann integrability is the same as being $L^1$ in the sense of Lebesgue, but if $f$ oscillates then the Lebesgue integral may not exist.

Further note that the uniform continuity is required to prevent sharp “spikes” that might prevent the limit from existing. For example suppose we add a spike of height 1 and area $2^{-n}$ at every integer. Then the function is continuous and $L^1$ (and thus Riemann integrable), but$f(t)$ would not have a limit at infinity.