We determine sufficient conditions under which certain recursively defined functions are well defined for all real inputs. Given a function $f:\mathbb R\to\mathbb R$, call a decreasing sequence $x_1>x_2>x_3>\cdots$ "$f$-bad" if $f(x_1)>f(x_2)>f(x_3)>\cdots$, and call the function $f$ "ordinal decreasing" if there exist no infinite $f$-bad sequences. We prove the following result: Given ordinal decreasing functions $f,g_1,\ldots,g_k,s$ that are everywhere larger than $0$, define the recursive algorithm "$M(x)$: if $x<0$ return $f(x)$, else return $g_1(-M(x-g_2(-M(x-\cdots-g_k(-M(x-s(x)))\cdots))))$". Then $M(x)$ halts and is ordinal decreasing for all $x \in \mathbb{R}$. The recursive algorithms $M$ and $M_n$ previously studied in the context of fusible numbers by Ericskon et al. (2022) and Bufetov et al. (2024), respectively, are special cases of this scheme. Moreover, given an ordinal decreasing function $f$, denote by $o(f)$ the ordinal height of the root of the tree of $f$-bad sequences. Then we prove that, for $k\ge 2$, the function $M(x)$ defined by the above algorithm satisfies $o(M)\le\varphi_{k-1}(γ+o(s)+1)$, where $γ$ is the smallest ordinal such that $\max\{o(s),o(f),o(g_1), \ldots, o(g_k)\} <\varphi_{k-1}(γ)$.

翻译：我们确定了某些递归定义函数对所有实数输入均有定义的充分条件。给定函数 $f:\mathbb R\to\mathbb R$，若递减序列 $x_1>x_2>x_3>\cdots$ 满足 $f(x_1)>f(x_2)>f(x_3)>\cdots$，则称其为“$f$-坏序列”；若不存在无限$f$-坏序列，则称函数$f$为“序数递减”函数。我们证明以下结果：给定处处大于$0$的序数递减函数 $f,g_1,\ldots,g_k,s$，定义递归算法“$M(x)$：若 $x<0$ 返回 $f(x)$，否则返回 $g_1(-M(x-g_2(-M(x-\cdots-g_k(-M(x-s(x)))\cdots))))$”。则对于所有 $x \in \mathbb{R}$，$M(x)$ 均会终止且为序数递减函数。Ericskon等人（2022年）与Bufetov等人（2024年）分别在可熔数研究中讨论的递归算法 $M$ 与 $M_n$ 均为本方案的特定情形。此外，给定序数递减函数 $f$，用 $o(f)$ 表示 $f$-坏序列树的根节点序数高度。我们证明对于 $k\ge 2$，由上述算法定义的函数 $M(x)$ 满足 $o(M)\le\varphi_{k-1}(γ+o(s)+1)$，其中 $γ$ 是满足 $\max\{o(s),o(f),o(g_1), \ldots, o(g_k)\} <\varphi_{k-1}(γ)$ 的最小序数。