Please someone tell me why $\max\{0,x\}$ is not a function at $x=0$. I always learned that to fail the vertical line test the function's graph should have different values for the same input. However, this function apparently has those different values equal to each other. Not coincidentally, the book shows a line there and says $max\{0,x\}$ is not a function at the point. I am really asking for help, not just being rhetorical.
Well by definition
$$\max\{0,x\}=\begin{cases} x, & x>0, \\ 0, & x\leq 0 \end{cases},$$
which is indeed a function $\mathbb{R}\to\mathbb{R}$. Your picture does not show this function, however, so I don't know how you made that connection.