WHat would be an example of this function?

Question

WHat would be an example of a Lebesgue integrable function $f:\mathbb{R}\rightarrow [0,\infty)$ satisfying:

1. $f$ is continuous.

2. $\limsup_{x\to \infty} f(x)=\infty$

Answer 1

Consider a continuous positive function $g$ with support $\subset [0,1]$, $\int_0^1g(x)dx<\infty$ and $g(1/2)=1$.

Then $f(x) = \sum_{n=1}^\infty g(n^3x - n^4)\times n$ is good:

• $\int g(x)dx = \sum \int g(n^3x - n^4)dx \times n = \int_0^1g(x)dx \sum n^{-2}<\infty $
• $g(n^4 + 1/2n^3) = f(1/2)\times n = n$ hence $\limsup f = \infty$.