Local versus global implicit function

Question

Suppose the equation $f\left(x,y\right)=0$, with $x\in I_{1}$ and $y\in I_{2}$, $I_{1}$ and $I_{2}$ being open intervals. Additionally, consider that the conditions required to apply the Implicit Function Theorem (IFT) are verified for all $\left(x_{0},y_{0}\right)\in I_{1}\times I_{2}$. Hence, we can conclude that in a neighborhood containing the point $\left(x_{0},y_{0}\right)$, the equation $f\left(x,y\right)=0$ defines implicitly $y$ as a function of $x$.

And my question is: Since the conditions of IFT hold for all $\left(x_{0},y_{0}\right)\in I_{1}\times I_{2}$, is it true that the equation $f\left(x,y\right)=0$ defines implicitly $y$ as a function of $x$ with the domain of this implicit function being $I_{1}$?

Answer 1

One of the conditions for the IFT to apply at $(x_0,y_0)$ is that $f(x_0,y_0)=0$. Thus, if we take what you write literally, the answer is that the conditions cannot be fulfilled on all of $I_1\times I_2$, since then $f(x,y)\equiv0$ on all of $I_1\times I_2$ and this function doesn't fulfill the requirement on the partial derivative anywhere.

This is probably not what you meant. Then the answer depends on what you did mean.

If you mean that the condition on the partial derivative is fulfilled wherever $f(x_0,y_0)=0$, then the answer is no. For instance, for $f:[0,1]\times [-2,2]$ with $f(x,y)=y^2-1$, there are two different implicit functions, $g(x)=1$ and $g(x)=-1$, and thus no global function.

If you mean that the condition on the partial derivative is fulfilled everywhere but $f(x_0,y_0)=0$ is only fulfilled somwhere, then the answer is still no, since there is no guarantee that for a given value of $x\in I_1$ there is any value of $y\in I_2$ such that $f(x,y)=0$.

However, under the assumption that for each $x\in I_1$ there is at least one $y\in I_2$ such that $f(x,y)=0$, and that the condition on the partial derivative is fulfilled everywhere in $I_1\times I_2$, then in this rather restricted sense the answer is yes, since for given $x$ a continuously differentiable function can't take the value $0$ for two different values of $y$ without the partial derivative with respect to $y$ vanishing somewhere in between, so there is guaranteed to be exactly one value of $y\in I_2$ with $f(x,y)=0$ for each $x\in I_1$.