Graph of Real Bijection in Coordinate Plane intersects Horizontal Line at One Point
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Theorem
Let $f: \R \to \R$ be a real function which is bijective.
Let its graph be embedded in the Cartesian plane $\CC$:
Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on exactly one point $P = \tuple {a, b}$ where $b = \map f a$.
Proof
By definition, a bijection is a mapping which is both an injection and a surjection.
Let $\LL$ be a horizontal line through a point $b$ in the codomain of $f$.
From Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line:
- $\LL$ intersects the graph of $f$ on at least one point $P = \tuple {a, b}$ where $b = \map f a$.
From Graph of Real Injection in Coordinate Plane intersects Horizontal Line at most Once:
- $\LL$ intersects the graph of $f$ on at most one point $P = \tuple {a, b}$ where $b = \map f a$.
The result follows.
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Exercise $6 \ \text {(c)}$