# Approximation of positive right-continuous function

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.

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

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

• Sure one can! Define $D = \{(x,y) : y < f(x)\}$ and consider $D_t = \{(x,y) : (x + t y, y + t x) \in D\}$. Then it is not very difficult to see that $D_t = \{(x, y) : y < f_t(x)\}$ for a continuous, strictly decreasing function $f_t$, and $f_t(x)$ increases to $f(x)$ as $t \to 0^+$. Jun 14, 2021 at 9:12
• (I realise the above works for $f$ defined on all of $\mathbb R$, and needs an appropriate adjustment for $f : (0,\infty) \to (0,\infty)$.) Jun 14, 2021 at 9:17
Take $$f_n(x) =n\int_{x}^{x+1/n} f(t) dt-x/n$$
• On $(0,\infty)$, the $-x/n$ part is just for strict monotonicity Jun 14, 2021 at 11:20
• Is it possible to make $\{f_{n}\}$ a sequence of positive functions? Jun 21, 2021 at 18:37
• Of course, replace $-x/n$ to $+1/(nx+n)$ Jun 21, 2021 at 19:19