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：摘要：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“$f$-不良序列”；若不存在无限$f$-不良序列，则称函数$f$为“序数递减函数”。我们证明以下结论：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$M(x)$对所有$x \in \mathbb{R}$停机且是序数递减的。Ericskon等人(2022)与Bufetov等人(2024)在可融合数背景下分别研究的递归算法$M$和$M_n$均为该框架的特例。进一步地，给定序数递减函数$f$，记$o(f)$为$f$-不良序列树根节点的序数高度，则我们证明：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。