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](https://math.stackexchange.com/questions/1517134/convergence-of-generalized-inverses) 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!



Approximation of positive right-continuous function
Take $f\_n(x) =n\int\_{x}^{x+1/n} f(t) dt-x/n$


