Example of function in L2 but not in H1

Question

I am having difficulty finding a function that is an element of $ L^2(\mathbb{R}) $ but is not an element of $ H^1(\mathbb{R}) $ on the whole real line. Can anyone think of an example?

Answer 1

Hint: Every function in $H^1(\mathbb R)$ is continuous.

Answer 2

The $L^2$ function basically could be anything. All typical examples of functions you met so far are $L^2$ functions.

However, $H^1$ is something special. Try to recall the definition of $H^1$, it contains something about "derivative" right? So, an question for you. Take function $f(x)=1$ if $x\in (0,1)$ but $f(x)=0$ otherwise. Is it a differentiable function? is it a $L^2$ function?