Is $H^1(0,\infty) \subset C^0([0,\infty))$?

Question

Is it true that $H^1(0,\infty) \subset C^0([0,\infty))$ is a continuous embedding? How would I prove it?

I do know this holds for bounded domains in one dimension but here we have the half line.

Thanks

Answer 1

This is Theorem 8.2 in Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis:

Let $ u \in W^{1,p}(I)$ with $I$ a bounded or unbounded interval. Then there exists $\tilde u \in C(\bar I)$ with $\tilde u = u$ a.e. on $I$ and $$\tilde u(x) - \tilde u(y) = \int_y^x u'$$ for all $x, y \in \bar I$.