Determining if a given function is a closed loop Is there any method/way to determine if a given maths function in $x$ and $y$, when plotted with the help of coordinate-axes, forms a closed-loop? In other words, does it, all by itself, enclose some area?
For example,
$x^2+y^2=5$
forms a closed loop (a circle).
But $y = x^2$
does not form a closed loop (as it is a parabola, and is open from above).
 A: The circle $x^2+y^2-5=0$ and the parabola $y-x^2=0$ are special cases of plane algebraic curves $p(x,y)=0$ where $p$ is a polynomial in two variables, and those in turn are special cases of zero-sets or loci of $f(x,y)=0$ where $f$ is any function $f:\mathbb{R}^2\to\mathbb{R}$.
In general, you can check whether the locus is a “closed loop” by verifying that it is a compact, connected 1-manifold.
The “compact” part boils down to “bounded” (I’m assuming that $f$ is continuous, so the inverse image $f^{-1}(0)$ is closed.) You need to see that $f(x,y)=0$ forces $|x|$ and $|y|$ not to exceed some fixed bound $B$. This is often, but not always, relatively easy to do by studying the behavior of $f$ for fixed values of $x$ or $y$.
The “1-manifold” part is usually handled by something like the Implicit Function Theorem. If $f$ is differentiable and the derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ don’t simultaneously vanish whenever $f(x,y)=0$, then the conditions of the Implicit Function Theorem are met and the locus is locally one-dimensional, namely, a curve. Singularities need to be explored further to determine if they still describe a topological curve (they may or they may not.)
The “connected” part may be the hardest. There’s no simple general procedure to apply: you need to study the locus and check that you can travel from any point to any other point. Tools from algebraic topology are sometimes applicable, but usually it’s easier to find a specific reason given the specific nature of the given function $f$.
