Give an example of unbounded continuous function that is Lebesgue integrable on $\mathbb{R}$?

Question

I can think of an example like $f(x) = x^{-1/2}$ defined on interval $[0, 1]$, but cannot extend it to $\mathbb{R}$ and make it continuous.

Answer 1

Take a nonnegative smooth function $f\not\equiv 0$ supported on $[0,1]$ where $f(0) = 0 = f(1)$, then make copies of it centered at each $n\in\Bbb N$ where the height increases by a factor of $n$, but the support shrinks by $\frac{1}{n^3}$, i.e. something like $nf(n^3x)$.

Answer 2

For $n\in \mathbb N$ let $A_n=[x_n,y_n]=[n-(n+1)^{-3}, n+(n+1)^{-3}].$

Let $f(x)=0$ for $x\in \mathbb R$ \ $\cup_{n\in \mathbb N}A_n.$

For $n\in \mathbb N$ let $f$ be linear on $[x_n,n]$ and let $f$ be linear on $[n, y_n]$ with $f(x_n)=f(y_n)=0$ and $f(n)=n.$

The details of showing that $f$ is Lebesgue-integrable over $\mathbb R$ are a bit "fussy" but entirely elementary. Use the fact that $\sum_{n\in \mathbb N}n^{-2}<\infty.$