Limit of a function defined implicitly

Question

Let a function $y:\mathbb{R} \setminus \left\{ 0\right\} \to \mathbb{R}$ be defined implicitly by the equation $F(y(x),x)=0$ for some $F:\mathbb{R} \times ( \mathbb{R} \setminus \left\{ 0\right\} ) \to \mathbb{R}$.

Assume that

1. $y$ is well defined, i.e., for each $x \in \mathbb{R} \setminus \left\{ 0\right\}$ there is a unique solution $y(x)$ of the equation $F(y(x),x)=0$.
2. The limit $c:=\lim_{x\rightarrow 0} y(x)$ exists.

Under what minimal conditions one might expect that $\lim_{x\rightarrow 0} F(c,x)=0$?

Answer 1

Hint:

From the condition 1., we draw that there is a function $\phi$ such that

$$F(x,y)=0\iff y=\phi(x).$$

So you are essentially asking what it takes to have

$$\lim_{x\to c}\phi(x)=\phi(\lim_{x\to c}x).$$