I consider this problem.
Supposed a continuous map $f:[0,\infty) \to \mathbb R$ and $A \in \mathbb R$
I want to prove that $$\lim_{n \to \infty}\frac {\int_0^n f(x) \, dx}{n}=A\quad\Longrightarrow\quad \lim_{x \to \infty}f(x) = A$$
Please tell me your idea.
This is true if you reverse the direction of the arrow: $$ \lim_{n\,\to\,\infty}\frac {\int_0^n f(x) \, dx} n = A \quad\Longleftarrow\quad \lim_{x\,\to\,\infty} f(x) = A. $$ An $\text{“ }\varepsilon\text{-}N\text{ ''}$ proof of that is not hard.
As remarked in comments, among functions that oscillate between two values, one can find counterexamples. In particular, for any periodic function that oscillates between two finite bounds, the average value of the function over one period will be the limit.