Real numbers subscript + and bar overhead

Question

Please refer to Lawrence Evans Book on PDE (2nd edition) page 70. http://www.mif.vu.lt/~algirdas/Evans.pdf (page 80 here) on the Euler-Poisson-Darboux equation. What does the notation $\bar{\mathbb{R}}_{+}$ mean? And why is $U\in C^{m}(\bar{\mathbb{R}}_{+}\times[0,\infty))$? How does that come from the fact that $u\in C^{m}(\mathbb{R}^{n}\times[0,\infty))$?

Answer 1

$$\mathbb{R}_+=[0,+\infty)\qquad\bar{\mathbb{R}}_{+}=\mathbb R_+\cup\{+\infty\}$$