Let $f:(0, +\infty)\to(0, +\infty) $ be a **monotone decreasing**, **right-continuous** function. Can we find a sequence $\{f_{n}\}_{n\in \mathbb{N}}$ of **strictly monotone decreasing**, **continuous functions**, such that $f_{n}$ converges pointwise to $f$, that is, $f(x)=\lim_{n\to \infty}f_{n}(x)$ for all $x\in (0, +\infty)?$ To achieve strict monotonicity is trivial, but I have problems with the continuity. I came across this question while reading this about convergence of generalized inverse functions.

[Edit]: Additional question: Is it possible to also make $\{f_{n}\}$ a sequence of positive functions?

[Edit]: Additional question two: Is it possible to make $\{f_{n}\}$ an increasing sequence of positive functions?

Thanks in advance for your help!