Trying to find a reparameterization of a function from $f(y, z, \ldots)(x)$ to $f(y(a_1), z(a_2), \ldots)(x)$ so that there exists some $C_0$ such that for all $x \in [r, t]$ we have $$ |f(y(a_1), z(a_2), \dots)(x) - f(y(b_1), z(b_2), \dots)| \le C\|a - b\|_1. $$ It seems similar to a bounded operator from functional analysis if you look at it as mapping from the vector $a$ to the function $f(\dots)$. With the first space having the $L^1$ norm and the second the $\sup$ norm. I haven't been able to find anything based on that googling though. Since it doesn't require it be linear just have norm bounded by a linear function.

I'm pretty sure there are functions that can't be parameterized this way, but I'm interested in being able to identify them and construct these reparameterizations.

For instance one I've had trouble proving if it works is $f(a, b)(x) = \frac{a}{1 + e^{-bx}}$.