I am struggling on the following problem and I couldn't find a reference to answer that question.
Let $F$ be a function of two variables such that $F\in\mathscr{C}^{1, \infty}(\mathbb{R}_+\times\mathbb{R},\mathbb{R})$. The function $F$ is bounded and also Lipschitz w.r.t the two variables and we have that:
$$\forall x\in\mathbb{R}_+,\exists!y_x\in\mathbb{R}\text{ s.t., } F(x,y_x)=0.$$
Can we say under these strong assumptions that the implicit function mapping $x$ to $y_x$ is Lipschitz ?
Otherwize, what condition could be added to make it true ?
The problem behind:
I have a function $f_a$ depending on a positive real parameter $a$ that has only one root $x_a$. I want to prove that for all $a_1,a_2>0$, there exists a constant $K$ such that :
$$|x_{a_1}-x_{a_2}|\leq K|a_1-a_2|.$$
Thank you very much!