Is continuous $L_2(\mathbb R)$ functions are in $L_1(\mathbb{R})$?

Question

Is continuous, $L_2(\mathbb R)$ functions are in $L_1(\mathbb{R})$?

When working on some problem I across this question. I was thinking that it is not true.

But I am not able to give example for that.

Even I am not able to prove it !

Please help me !

Answer 1

$$ \frac{1}{\sqrt{1+x^2}} \in L^2\setminus L^1. $$ And this function is continuous everywhere. So the answer to your question is "No."

Answer 2

Let $f(x)=1/|x|$ for $|x|\geq 1$ and $f(x)=1$ for $|x|\leq 1$, then $f\in L^{2}({\bf{R}})$ but $f\notin L^{1}({\bf{R}})$, and $f$ is continuous throughout.