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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!

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