Suppose $f\in H^2(\mathbb{R})$ with $\lVert f\rVert_{H^1}<\varepsilon$.
Does this imply that $f\in C^1(\mathbb{R})$?
I know that for open and bounded $\Omega\subset\mathbb{R}$, one can embed $H^2(\Omega)$ into $C^1(\Omega)$.
But I guess this is not true for $\Omega=\mathbb{R}$.
The Sobolev spaces on $\mathbb R$ are actually easier to study than the ones on domains, because you can use the Fourier transform. The $H^2$ norm can be defined to be
$$
\lVert f\rVert_{H^2}=\sqrt{\int_{-\infty}^{\infty} \lvert \hat{f}(k)\rvert^2(1+k^2)^2\, dk}, $$
and the Fourier inversion formula, which reads
$$
f(x)=\int_{-\infty}^\infty \hat f(k)e^{ixk}\,dk, $$
implies, via Cauchy-Schwarz, the inequality
$$\tag{1}
\lvert f(x)\rvert \le C\lVert f\rVert_{H^2},\quad \forall x\in \mathbb R,$$
with $C=\sqrt{ \int_{-\infty}^{\infty} (1+k^2)^{-2}\,dk}$. This is the embedding $H^2(\mathbb R)\subset C(\mathbb R)$.
I now read your question more carefully and realised you wanted the embedding $H^2(\mathbb R)\subset C^1(\mathbb R)$. This is obtained the same way, starting with $$f’(x)=i\int_{-\infty}^{\infty} k\hat f(k) e^{ikx}\, dk,$$ yielding the inequality $$\tag{2} \lvert f’(x)\rvert \le C_1 \lVert f\rVert_{H^2}, $$ with $C_1=\int_{-\infty}^{\infty} k^2(1+k^2)^{-2}\, dk$. The inequalities (1) and (2) are the desired embedding $H^2(\mathbb R)\subset C^1(\mathbb R)$.
