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 ?

What about $\mathrm{f}(x)=T$?
$$ \lim_{T\to\infty}  \frac{1}{T}\, \int_0^T \mathrm{f}(x)~\mathrm{d}x=\lim_{T\to\infty}  \frac{1}{T}\, \int_0^T T~\mathrm{d}x=\lim_{T\to\infty}  \frac{1}{T}\, T^2=\infty$$ 
