Suppose $F: \mathbb{R} \rightarrow \mathbb{R}$ is continuous everywhere. Is it true that the limit
$\lim_{n \rightarrow \infty} \int_{-n}^n f(x) \; dx$
always exists?
Suppose $F: \mathbb{R} \rightarrow \mathbb{R}$ is continuous everywhere. Is it true that the limit
$\lim_{n \rightarrow \infty} \int_{-n}^n f(x) \; dx$
always exists?
No. You could have an oscillatory function such as $f(x)=\cos(\pi x/2)$ with an oscillatory integral: $\sin(\pi n/2)-\sin(-\pi n/2)$ does not tend to a limit as $n \to \infty$ since it's 0 for even $n$ and 2 for odd $n$.
No. It's not true. The integral of the constant function will converge for any finite n, but the limit does not, for example.