Let $A\subset \mathbb R^n$ be open and $f : A \to \mathbb R^n$ is class $C^1$, then there exists $M>0$,such that for all $x \in A, y \in \mathbb R^n$
$$\Vert Df(x)y\Vert\le M\Vert y\Vert$$
This is a part of proof of Inverse function theorem that i don't get.
Obiously every linear map on $\mathbb R^n$ is bounded so $$(\forall x \in A)(\forall y \in \mathbb R^n)(\exists M_x>0)(\Vert Df(x)y\Vert \le M_x\Vert y\Vert)$$
However if we look at the $M:= \sup_{x\in A}M_x $, I dont see the argument that $M < \infty$
Edit: In the original text, i am only looking at $K(0,r) \subseteq A$ where $\det Df(x) \not = 0$