Does the improper integral exist? I need to find a continuous and bounded function $\mathrm{f}(x)$ such that the limit
$$ \lim_{T\to\infty}  \frac{1}{T}\, \int_0^T \mathrm{f}(x)~\mathrm{d}x$$ 
doesn't exist. 
I thought about $\mathrm{f}(x) = \sin x$ but I am not sure if the fact that we divide by $T$ may some how make it converge to zero.
What do you think ?

The integral has to diverge to infinity if this limit is to not exist, (of course any other limit will give zero overall). Given the type $``\dfrac{\infty}{\infty}"$ limit, if we apply L'Hopitals; $$\displaystyle\lim_{T\to \infty}\dfrac{1}{T}\displaystyle\int_{0}^{T}f(x)\ dx = \lim_{T\to \infty} f(T)$$
This makes it clear that the limit doesn't exist iff $f$ is not bounded, so there is no suitable function. 
