How prove this function have bounded and such $\lim_{r\to\infty}\frac{\int_{-r}^{r}|f(x)|dx}{2r}=k$

Question

Question:

Let $f$ is continuous on $R$,and such $$\lim_{r\to+\infty}\dfrac{\int_{-r}^{r}|f(x)|dx}{2r}=k$$

where $k$ Can be infinite.

prove or disprove $f(x)$ is Bounded function?

My try: I think we can consider follow two case:

case 1: if $k<+\infty$,

case2: if $k=+\infty$

can you someone can take comtexample,Thank you very much!

Answer 1

Let $f(x) = 0$ except where it has a continuous "bump" with width $\frac{1}{n}$ and height $n$ (and so integral $\leq 1$) whenever $n = 2^m$ is a positive integer power of two. Then $f(x)$ is not a bounded function, but the limit in question is $0$. So $f(x)$ need not be bounded in general for any $k$.