# Constant of continuity as a monotonically increasing function of the radius of the neighbourhood

While reading the fairly famous paper by Brezzi et al from 1980 (link), I got slightly puzzled by one of the conditions, in particular in Theorem 1 (Implicit Function Theorem basically). They consider a function $$F \in C^1(X\times Y,Z)$$ and in many places require conditions such as: there exists a monotonically increasing function $$L_1\,:\,\mathbb{R}_+ \to \mathbb{R}_+$$ such that for all $$(x,y) \in S(x_0,y_0,\xi)$$, where $$S(x_0,y_0,\xi) := \{(x,y) \in X \times Y\,|\, \|x-x_0\|_X + \|y-y_0\|_Y \leq \xi\}$$, it holds that $$\|Df(x,y) - Df(x_0,y_0)\| \leq L_1(\xi)(\|x-x_0\|_X + \|y-y_0\|_Y).$$ It is fairly obvious from the proofs that when they mean 'monotonically increasing', they do not require 'strictly increasing', hence $$L_1(\xi) = C$$ (constant) would do (unless I'm mistaken?). Now, since $$f$$ is assumed to be $$C^1$$, I guess a constant would not work because $$DF$$ is simply continuous, but not uniformly continuous. Would it be always be true if, say, $$f \in C^2$$, as then we can argue that $$\|Df(x,y) - Df(x_0,y_0)\| \leq \|D^2f(x_*,y_*)\|(\|x-x_0\|_X + \|y-y_0\|_Y),$$ where $$(x_*,y_*) \in S(x_0,y_0,\xi)$$? Are there any subtleties here that I fail to grasp? Is it a strong assumption? I will greatly appreciate any comments, thanks!