The vertical line test

($\Rightarrow$) Suppose $C$ is the graph of a function $f$, so every point of $C$ has the form $(x_0,f(x_0))$ for some input $x_0$. Fix any vertical line $x=x_0$; the only points of $C$ that can lie on it are those with first coordinate $x_0$, and there is only one such point, $(x_0,f(x_0))$, because $f$ assigns a single value to $x_0$. So the line meets $C$ in at most one point.

($\Leftarrow$) Conversely, suppose every vertical line meets $C$ in at most one point. Define $f$ by: for each $x_0$ that has a point of $C$ above it, let $f(x_0)$ be the $y$-coordinate of that (unique, by hypothesis) point. This assigns at most one output to each input, so $f$ is a well-defined function, and by construction its graph is exactly $C$.

The two directions together show the equivalence: being a function's graph and passing the vertical line test are the same property viewed from two angles.