Name of a concept in math analysis in English

[nuuskur]

The concept's main theorem goes as follows:
Suppose that $w = F(x,y)$ satisfies the following criteria at point A = (a,b).
$$\begin{cases}\exists\delta > 0\colon F\ is\ continuous \forall (x,y)\in B(A,\delta)\\<br /> \exists \frac{\partial}{\partial y}F , continuous \\<br /> F(A) = 0, F_y \neq 0\end{cases} \Rightarrow \exists y = f(x) continous$$

The absolute correct definition is an utter pain to get written down, though that's not my purpose. This is enough information to be able to recognize what I am getting at. We call these things "undeveloped" or "inexplicit" functions, but I can't find anything with those keywords so it's (again) my apparent lack of vocabulary that's at work here :<

 

[Mark44]

The concept's main theorem goes as follows:
Suppose that $w = F(x,y)$ satisfies the following criteria at point A = (a,b).
$$\begin{cases}\exists\delta > 0\colon F\ is\ continuous \forall (x,y)\in B(A,\delta)\\<br /> \exists \frac{\partial}{\partial y}F , continuous \\<br /> F(A) = 0, F_y \neq 0\end{cases} \Rightarrow \exists y = f(x) continous$$

The absolute correct definition is an utter pain to get written down, though that's not my purpose. This is enough information to be able to recognize what I am getting at. We call these things "undeveloped" or "inexplicit" functions, but I can't find anything with those keywords so it's (again) my apparent lack of vocabulary that's at work here :<

I think this is the Implicit Function Theorem. In English, the opposite of "explicit" is "implicit", not "inexplicit."

 

[nuuskur]

much appreciated.