As the implicit function theorem shows, if

(i)Function F is continuous in the region D$\subseteq R^2$;

(ii)F($x_0,y_0)=0,P_0(x_0,y_0)\in$D;

(iii)There is a continuous partial derivative $F_y$(x,y)=0 in the region D;

(iv)$F_y(x_0,y_0)\neq$0;

Then there uniquely exists the function y=f(x) defined in the interval ($x_0-\alpha$,$x_0+\alpha$),that

1 f(x$_0$)=$y_0$,(x,f(x))$\in$U(P$_0$)when x$\in$(x$_0$-$\alpha$,x$_0$+$\alpha$)and F(x,f(x))$\equiv$0;

2 f(x) is continuous in ($X_0-\alpha$,$X_0+\alpha$).

And what if the conditions are weaker than those above? That is (i)Function F is continuous in the region D$\subseteq R^2$;

(ii)F($x_0,y_0)=0,P_0(x_0,y_0)\in$D;

(iii)There is a continuous partial derivative $F_y$(x,y)=0 in the region D;

In other words,what is the conclusion for the existence of implicit function in the branch or the subset of $R^2$?

There have been a series conclusions in complex space，who can list some of them and what's the relationship with the situation of complex space and real space?