Prove that a function is $L^p(\mathbb{R})$

Question

There is a specific criterion for proving that a function $f \in L^p(\mathbb{R})$ as well as proving it by definition ?

Furthermore, is correct to imply that: If $|\ f|^{\ p}$ is continuous in $\mathbb{R}$, and $$\lim_{t \to \pm \infty} |\ f(t)\ |^{\ p} = 0 $$ Then $f \in L^p(\mathbb{R})$ ?

Answer 1

1. Basically everything comes down to the definition. There are various ways of proving that an improper integral converges.
2. Does the improper integral of a continuous function converge if the integrand goes to $0$ at $\infty$?

Answer 2

No. Positively not. A continuous function on $\Bbb R$ that tends to zero at infinity need not be in $L^p(\Bbb R)$; saying it's in $L^p$ says something about how fast it tends to infinity.

For example consider $$f(t)=\frac1{(1+|t|)^{1/p}}.$$

On a related note, a continuous function that lies in $L^p(\Bbb R)$ need not tend to zero at infinity, although here I don't think one can write down a single simple formula for an example.

Answer 3

Adding to David C. Ullrich's answer, we can construct a continuous $L^p$ function which doesn't vanish at infinity as follows. For each $n$, let $f_n$ be $0$ outside of $(n,n+2^{-n})$, $1$ at $n+2^{-(n+1)}$ and linearly interpolated in between. Explicitly, we have $$f_n(x)=[1-2^{n+1}|n+2^{-(n+1)}-x|]\mathbf 1_{(n,n+2^{-n})}(x).$$ Now let $f(x):=\sum_{n=1}^\infty f_n(x)$. It is not hard to show that this series converges pointwise to a continuous function and $f\in L^p(\mathbb R)$ for all $1\le p\le\infty$, however $\lim_{x\to\infty}f(x)$ does not exist.