Give an example of $h$ and $h_n$ where h is not continuous at $0$.

Question

All of the following have to hold for the function and the sequence that are given as an example:

1) $h(x)=lim_{n\rightarrow\infty}h_n(x)$

2) $\{h_n(x)\}$ is a sequence of decreasing non-negative functions on $\mathbb{R}$.

3) h(x) cannot be continuous at $0$.

Any tip or hints at building this example will be greatly appreciated!

Answer 1

Take $h_n(x)=h(x)=e^{-x}+1$ for $x <0$, $h_n(x)=h(x)=e^{-x}$ for $x \geq 0$.