TITLE: Can a 'closed' curve be a function?
QUESTION [8 upvotes]: For a 'closed' curve (I don't know if there is such in mathematics called closed curve, but I mean a curve which is 'closed')
e.g.1)It is well-known unit circle with equation$$x^2+y^2=1$$e.g.2)It is the function $f$ which is in my another post.I observed that they cannot be a function, since one $x$-value may correspond to several $y$-values, or will there be a 'closed' curve, such that one $x$-value can only correspond to at most one $y$-value, i.e., it is a function?

REPLY [2 votes]: A simple closed curve can't be the graph of a 1-variable function on $\mathbb{R}$. By the Jordan curve theorem, a closed curve partitions its complement into two parts, a bounded interior and an unbounded exterior. Pick any point in the interior and consider the vertical line through it. Since the interior is bounded both the ray on the line above the point and the ray below the point must leave the interior. The points where they leave correspond to failures of the vertical line test, hence the curve can't be the graph of a function.
Note that in the above proof, the vertical direction is somewhat arbitrary. The same basic argument shows that a simple closed curve can never be the graph of a function even after a rotation of axes. This contrasts with e.g. the case of parabolas. They can fail to be the graph of a function with respect to one choice of coordinate system , but you can always rotate the coordinate system so that the axis of symmetry of the parabola is parallel to the new y-axis (hence is the graph of a function with respect to that coordinate system). The fact that a simple closed curve is not the graph of a function is independent of coordinate system (you can clearly also translate the origin without changing anything essential in the argument). 
Question: is the Jordan curve theorem equivalent to the statement that a simple closed curve is not the graph of a function with respect to any choice of coordinate system? If you knew this fact about simple closed curves, could you use it to prove the JCT?