Theorem 5.4 in the book Concentration Inequalities: A Non-Asymptotic Theory of Independence by Boucheron, Lugosi and Massart, has the following environment:
Let $X$ be a standard normal random variable and let $f: \mathbb{R} \to \mathbb{R}$ be a continuously differentiable function.
I do not believe it is important to know what the Theorem is about to answer this question, only that it has the environment above.
In the proof of their Theorem, the authors argue "by a standard density argument" that if $E \left[f'(X)^{2} \right]<\infty$ then "it suffices to prove the theorem for twice differentiable functions with bounded support."
My question: Is there a connection between square-integrability of the first derivative and second-order differentiability? In addition, does square integrability of the first derivative imply that the function $f(x)$ tends to $0$ as $x \to \pm \infty$?
I am really looking for help on what this "standard density" argument means, and why it allows us to consider twice differentiable functions with bounded support (rather than continuously differentiable functions with unbounded support) when $E \left[f'(X)^{2} \right]<\infty$.