Let be $$g_n(x)=\begin{cases} ~0 & \text{if } x \leq 0 \\ ~1 & \text{if } x=\frac{1}{n}\\ ~0 & \text{else} \end{cases}%$$
$g$ is affine and continuous if $x\leq 0$: $g_n(x)$ is constant and equals to $0$.
if $x\geq 0$ then let be $N$ such that $\frac{2}{N}\leq x$ then the sequence $g_n(x)$ is constant and equal to $0$ from some rank $N$ thus the sequence is constant and equal to $0$ from a given rank $N$.
Thus $g_n(x)\rightarrow 0, \forall x\in \ \mathbb{R}$
I'm not quite sure to understand the passage in italic. Indeed it goes without saying that from a given rank $N$, as far as the third case of the equation system is $0$ when it goes beyond $\frac{1}{N}$
