Regularity of function in the Sobolev space $H^1(\mathbb{R})$

Question

Let us define the Sobolev space $H^1(\mathbb{R})$ as the closure of $\mathscr{C_c}^1(\mathbb{R})$ (the continuously differentiable functions on $\mathbb{R}$ with a compact support) for the norm $\int_\mathbb{R} (f^2 + f'^2)$.

I have two questions.

1. It is well-known that for all function in $H^1(\mathbb{R})$, there exist a continuous representative. Can you confirm me that the functions of $H^1(\mathbb{R})$ are not necessarily continuous but the representative tends to $0$ at $\mp \infty$.

2. It seems to be true that if $f$ is in $H^1(\mathbb{R})$, $f^2$ is also in $H^1(\mathbb{R})$ but I am surprised that there is not any reference (a book for example) about it.

Answer 1

1. See here
2. Let $f(x)=|x|^{-a}$, modified in such a way that it goes to $0$ at $\infty$ fast enough (so that we only need to worry about what happens at $x=0$). With the definition of $H^1$ we know $f \in H^1\iff a<1$. So, if you take $a=3/4$, then $f\in H^1$ but $f^2 \notin H^1$