Is an elliptic curve a function? I am currently reading the book Analysis 1 from Terence Tao. The way he defines a function says that it should pass the vertical line test. Sometimes ago I came across an elliptic curve and saw that it doesn't pass the vertical line test. Is the elliptic curve a function ?
 A: It's not the graph of a function, there is no $f:\mathbb{R} \mapsto \mathbb{R}$ such that an elliptic curve is described by the set $\{(x, f(x)): x \in \mathbb{R}\}$, but if you can parametrize it with some $\gamma: [0, 1] \mapsto \mathbb{R}^2$ you can see it as the image of a function.
A: No, an elliptic curve such as $y^2=x^3$ is not a function. Even if we restrict its domain to $x \ge 0$, there are two points $(x, \pm y)$ on the curve for each $x > 0$. Similarly a circle such as $x^2 + y^2=1$ is not a function. The correct description for such curves is an algebraic variety.
A: No, it's not a function. The definition of an elliptic curve is the set of all points such that $y^2 = x^3 + ax + b$. That is, $E_{a, b, c} \equiv \{ (x, y) : y^2 = x^3 + ax + b \}$.
Let a solution $(x_0, y_0) \in E_{a, b, c}$. then by definition $y_0^2 = x_0^3 + a x_0 + b$,. So, $(-y_0)^2 = y_0^2 =  x_0^3 + ax_0 + b$. Therefore, $(x_0, -y_0) \in E_{a, b, c}.$ Hence, this will fail the vertical line test at $x = x_0$.
That is to say, an elliptic curve cannot be a function.
Some pictures of elliptic curves to see the symmetry along the $x$-axis:

