In his book, Differential Equations Demystified, Steven G. Krantz writes that the unit circle can be expressed: $$y=+\sqrt{1-x^2}$$ when $y$ is positive and $$y=-\sqrt{1-x^2}$$ when $y$ is negative. He adds: "It is only at the exceptional points $(-1,0)$ and $(-1,0)$ where the tangent lines are vertical that $y$ cannot be expressed as a function of $x$."
First off, let me say that I think he means to identify those points as $(-1,0)$ and $(1,0)$, but even allowing for the typo, I'm still a wee bit mystified.
Either of those two points represents a solution to both equations. Is it that there is the equations give the same value for $y$ at $x=0$? What is his point here?
Take a look at the following picture:
If we look near enough of $x$, it is clear that this picture represents a function: we have associated to every $x$ a unique $y$. For instance:
But if we slide $x$ to the right part, this does not hold anymore: we have $2$ values associated to $x$, no matter what neighbourhood of $x$ we choose:
But if we also restrict ourselves on a neighbourhood of one of those values associated to $x$, we can arrive at $y$ as a function of $x$. For example:
However, if $x$ is at the far right, there will be no way to do this, no matter what neighbourhood of $y$ and $x$ you take! See the following picture:
This occurs due to the fact that the picture is "vertical" at the far right. This issue is adressed formally by the Implicit Function Theorem.