Let $V$ be a neighborhood of the origin in $\mathbb R^2$ and $f: V\rightarrow \mathbb R$ a $C'$ map such that $f(0,0)=0$. Assume $f(x,y)\ge 2y$ for $(x,y)\in V$. Show that there is a neighborhood $U$ of the origin in $\mathbb R^2$ and a constant $L$ such that if $(x_1,y_1),(x_2,y_2)\in U$ and $f(x_1,y_1)=f(x_2,y_2)=0$, then $|y_2-y_1|\le L |x_2-x_1|$.
The idea is to apply the implicit function theorem to the map $f: V\subset \mathbb R^2\rightarrow \mathbb R$ such that $f(0,0)=0$. Since $f(x,y)\ge 2y$, $\partial f/\partial y\ne0$ at $(0,0)$. Thus by the IFT there is a neighborhood $W\subset \mathbb R$ of $0$ on the $x$-axis and a $C'$ map $g: W\rightarrow \mathbb R$ such that $g(0)=0$ and for any $t\in W$, $f(g(t),t)=0$.
But how to get hold of a neighborhood $(U)$ of the origin in $\mathbb R^2$ with the condition needed?
HINT: Apply the multi-variable mean value theorem.