Graph of Real Function in Cartesian Plane intersects Vertical at One Point
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Theorem
Let $f: \R \to \R$ be a real function.
Let its graph be embedded in the Cartesian plane $\CC$:
Every vertical line through a point $a$ in the domain of $f$ intersects the graph of $f$ at exactly one point $P = \tuple {a, \map f a}$.
Proof
From Equation of Vertical Line, a vertical line in $\CC$ through the point $\tuple {a, 0}$ on the $x$-axis has an equation $x = a$.
A real function is by definition a mapping.
Hence:
- $\forall a_1, a_2 \in \Dom f: a_1 = a_2 \implies \map f {a_1} = \map f {a_2}$
where $\Dom f$ denotes the domain of $f$.
Thus for each $a \in \Dom f$ there exists exactly one ordered pair $\tuple {a, y}$ such that $y = \map f a$.
That is, there is exactly one point on $x = a$ which is also on the graph of $f$.
The result follows.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions