A continuous integrable function over $[0,\infty)$ is not necessarily tending to $0$ at infinity. And a continuous functions tending to $0$ at infinity is not necessarily integrable. I think we can put both these classes of function in larger one, namely the set of continuous functions $f:[0,\infty)\to \mathbb{R}$ with vanishing mean i.e. $$\lim_{x\to\infty}\frac{1}{x}\int_0^x|f(s)|ds=0.$$
Can I find somewhere a reference which study this kind of functions and their properties?