TITLE: Why can a circle be described by an equation but not by a function?
QUESTION [27 upvotes]: In high-school math functions always looked to me just like glorified equations. The only time I saw a meaningful difference was when we covered the equation of a circle and I realized that an equation can describe a circle but a function cannot (a function can describe only one half of the circle).
So what is the difference between a function and an equation that leads to this, in formal terms?

REPLY [3 votes]: If you allow functions to misbehave a bit (multi-valued functions) like square roots for example, then a circle can be described as a function. (e.g. the square root of 9 is both +3 and -3 (since both values, when squared, yield 9)). 
E.g. solving $x^2 + y^2 = r^2$ for y:
$y = \sqrt{r^2 - x^2}$
Outside the circle (i.e. when x < -r or x > r), y is undefined, but within the circle, the square root above has two roots forming the top and bottom parts of the circle. 
Hmmm - I don't have enough rep to comment, hence the answer.